evaluation of certain infinite series using ......uation of other particular infinite series,...
TRANSCRIPT
EVALUATION OF CERTAIN
INFINITE SERIES
USING THEOREMS
OF JOHN, RADEMACHER
AND KRONECKER
by
Colette Sharon Haley
B .A . (Honours), Carleton University
A thesis subm itted to
the Faculty of Graduate Studies and Research
in pa rtia l fu lfilm ent of
the requirements fo r the degree of
Master of Science
School of Mathematics and Statistics
Ottawa-Carleton In s titu te for Mathematics and Statistics
Carleton University
Ottawa, Ontario, Canada
© C opyright
2004
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A bstract
John’s theorem relates the sum o f a certain in fin ite series involving a real
valued periodic function of period 1, which is o f bounded variation on [0, 1],
to a Riemann integral. A detailed proof of th is theorem based on the proof
o f Rademacher is given. John’s theorem is then used to determine the sum
o f some interesting in fin ite series. For example i t is shown th a t the sum of
1 _ 1 2 _ 2 2 _ 2 3 _ 3 £ _ _ 3 _ _3___3_ _ ± _ ±2 3 + 4 5 + 6 7 + 8 9 + 10 11+ 12 13+ 14 15+ 16 17 + " '
is Euler’s constant, see Theorem 1.6.2. Rademacher’s theorem is the exten
sion o f John’s theorem to algebraic number fields and th is theorem is applied
to determine the sum o f fu rther in fin ite series such as
1 1 1 2 1 1 2 7r log 21 2 4 + 5 8 + 9 10 + 4 ’
see Theorem 2.4.2. F inally, Kronecker’s lim it form ula is used to determine
the sums o f the in fin ite series
“ ( - 1)™ ~ ( - 1 )n “ ( - 1)™+"am? + bmn + cn? ’ ^ am2 + bmn + cn? ’ am2 + bmn + cn? ’
77i,n=—oo m ,n = —oo m ,n = —oo(m ,n )^ (0 ,0 ) (m ,n )^ (0 ,0 ) (m ,n ) / ( 0 ,0 )
fo r any prim itive , positive-definite, integral, b inary quadratic form
ax2 + bxy + cy2.
i
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Acknowledgem ents
F irs t and foremost, I would like to thank my supervisor D r. Kenneth S.
W illiam s. He helped me to choose a topic and has guided me through the
d ifficu lt task of w riting a thesis. He has also been remarkably patient and
understanding, and for th is I thank him.
I would like to acknowledge the help and support from friends who have
encouraged me along the way, especially Chad Ternent, Tom Maloley and
M a tt Lemire.
I have dealt w ith many computer problems recently and would like to
thank everyone who has helped me to solve them.
This thesis was w ritten in George Gratzer’s book “M ath into
has been an extremely useful reference.
Lastly I am solely responsible for any errors and shortcomings le ft in th is
thesis.
ii
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N otation
Z = domain o f integers
N = set of positive integers
Q = fie ld o f rational numbers
R = fie ld of real numbers
gcd(m, n) = greatest common divisor o f m, n 6 Z, (to, n) ^ (0,0)
( n 17 = Euler’s constant = 0.5772156649... = lim I V " ' - — log n
n —>oo \ ^ ' ?\ i= l
[rc] = greatest integer < x (x 6 R)
{ z } = fractional part o f x = x — [x] (x G R)
[a, b] = {x € R |a < x < b} (a, b € R, a < b )
{xo, X i, . . . , $„} = partition of [a, 6] given by a — xq < X\ < • • • < xn = b
A x k - x k - x k- \ (k = 1, 2 ,. . . , n)
p[a, 6] = set of all partitions of [o, 6]
= Legendre-Jacobi-Kronecker symbol (d € Z, d = 0,1 (mod 4) ,n
i i i
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H (d ) = form class group o f discrim inant d
h(d) = form class number = order of form class group H (d)
f = conductor of discrim inant d
OO 00
5 3 / ( m»n) = 5 3m , n - —oo m ,n = —oo
(m ,n )^ (0 ,0 )
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Contents
Abstract i
Acknowledgements ii
Notation iii
Introduction 1
1 John’s Theorem 5
1.1 Functions o f bounded variation .................................................. 5
1.2 John’s th e o re m ............................................................................... 17
1.3 P roof o f John’s th e o re m ............................................................... 23
1.4 Evaluation o f certain in fin ite series using John’s theorem . . . 36
1.5 The generalized Euler co n s ta n ts ...................................................... 41
v
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CONTENTS vi
1.6 Dr. Vacca’s series for 7 ......................................................................49
2 Rademacher’s Theorem 56
2.1 N o ta tio n ............................................................................................... 56
2.2 Rademacher’s extension of John’s
th e o re m ................................................................................................ 61
2.3 Rademacher’s theorem for algebraic
number f ie ld s .......................................................................................62
2.4 Rademacher’s theorem for im aginary
quadratic f ie ld s ................................................................................ 63
2.4.1 RT = 0 (7 = 1 ), c = 1 + z .................................................... 65
2.4.2 t f = Q (7 = 2 ), c = 7 = 2 ........................................................69
2.4.3 K = Q {V = 7 ) ,c = 1 + f = 7 ..................................................72
2.5 Rademacher’s theorem for real quadratic fie ld s .............................75
2.5.1 K = Q (V2), c = 7 2 .......................................................... 77
2.5.2 K = Q(^/p), P (prime) = 3 (mod 4 ) ....................................... 79
2.6 Rademacher’s theorem for real cubic fields w ith two nonreal
em bedd ings.......................................................................................... 81
2.6.1 K = Q (7 2 ), c = 7 2 .......................................................... 84
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CONTENTS vii
3 Kronecker’s Theorem 86
3.1 Dedekind eta function .................................................................... 86
3.2 Weber’s fu n c tio n s ............................................................................. 90
3.3 Kronecker’s lim it fo rm u la ................................................................ 91
3.4 F ina l re s u lts .........................................................................................103
Conclusion 109
Bibliography 110
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Introduction
The aim of th is thesis is to use three theorems, namely John’s theorem,
Rademacher’s extension of John’s theorem, and Kronecker’s lim it formula,
to determine the sums o f certain in fin ite series.
In 1934 F ritz John [20] showed th a t i f f ( x ) is a real-valued function defined
for a ll real x such tha t
(1) f ( x ) is periodic of period 1,
(2) f ( x ) is o f bounded variation for 0 < x < 1,
Vand c — - is a rational number > 1 w ith q > 0 and gcd(p, q) = 1, then the Q
in fin ite series
1
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INTRODUCTION 2
converges for a ll real values o f t, where
/
q, i i p ) ( n , q \ n ,
q - p , i f p | n, q [ n,
and its sum is
logc / f ( x ) dx.J o
Since f ( x ) is o f bounded variation on [0,1], f ( x ) is Riemann integrable on
[0,1], and so f * f ( x ) dx exists. In 1936 Hans Rademacher [27] proved John’s
theorem w ith (2) replaced by the weaker condition
(2)' f ( x ) is Riemann integrable on [0,1].
We give an exposition o f Rademacher’s proof o f John’s theorem expanding
on the details where necessary in Section 1.3.
Included in the volume [28] featuring his “lost” notebook are some frag
ments o f papers by Ramanujan. In particular, on pages 274 and 275 in [28],
there is the beginning o f a manuscript th a t probably was to focus on in te
grals related to Euler’s constant 7 . Berndt and Bowman [8] have presented
Ramanujan’s work in th is fragment and have related i t to other theorems in
the literature. In particu lar they prove [8, Lemma 2.5, p. 21] an integral
given by Ramanujan for Euler’s constant 7 , namely,
7 = /( 1 — xnn
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INTRODUCTION 3
and use it to obtain (among others) a series representation of 7 due to Vacca
[33], namely
I t is our purpose to show th a t Vacca’s form ula for 7 , as well as the eval
uation o f other particu lar in fin ite series, follows easily from John’s theorem.
We just mention one such evaluation. We show in Section 1.5 how Liang
and Todd’s evaluation [24] o f the in fin ite series where k is
a positive integer, can be deduced easily from John’s theorem.
In his 1936 paper, Rademacher also extended John’s theorem to algebraic
number fields. In Chapter 2 we use Rademacher’s theorem to obtain the
evaluation o f certain in fin ite series involving binary quadratic forms. In the
case of positive-definite forms, we show for example tha t
Rademacher’s theorem. In the case of indefinite forms we prove for example
tha t
1<|m W |l<3+2>/2I 771—n v 2 I
In Chapter 3 we use Kronecker’s lim it form ula to evaluate the in fin ite
series
(m ,n )^ (0 ,0 )
where the sum is ordered by increasing values of m 2+ m n + 2n2, follows from
OO
m ,n = —0 0 m + n y /2 > 0
oo f 1 \m
(m,ra)^(0,0)
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INTRODUCTION
E ( - i ) "am2 + bmn + cn21
771,71— —OO(m ,n )^ (0 ,0 )
^_-Qm+n
E am2 + bmn + cn2 ’771,71=—OO
(m ,n )# ( 0,0)
for any positive-definite, p rim itive , integral, binary quadratic form a
bxy + cy2. As an illus tra tion o f these, results, we show tha t
^ ( _ l ) m + n 47T ,
^ m 2 + 19n2 >/!9 g ’771,71=—OO v
(m ,n )^ (0 ,0 )
where 0 is the unique real root o f the cubic equation x3 — 2x — 2 = 0.
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Chapter 1
John’s Theorem
1.1 Functions of bounded variation
John’s theorem, which is the main topic o f th is chapter, expresses an in fin ite
series involving a periodic function of bounded variation as a Riemann in
tegral. We begin by reviewing the basic properties of functions o f bounded
variation. We follow the treatm ent given by Apostol in [4, pp. 165-169], see
also [30, pp. 117-121].
Throughout th is section, a and b denote real numbers w ith a < b .
D e fin itio n 1.1.1. Let f be a real-valued function defined on the closed in
terval [a, 6]. If, fo r every pa ir o f points x and y in [a, b], x < y implies
f i x ) < f ( y ) , then f is said to be increasing on [a, b\. I f x < y implies
f ( x ) < f ( y ) then f is said to be strictly increasing on [a, 6]. Decreasing and
5
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CHAPTER 1. JOHN’S THEOREM 6
strictly decreasing functions are sim ilarly defined.
Definition 1.1.2. A real-valued function f defined on the closed interval
[a, b] is said to be monotonic i f i t is increasing on [a, b] or decreasing on
[a,b].
Definition 1.1.3. I f [a,b] is a fin ite interval, then a fin ite set o f points
P = { x 0, x i , . . . , x n}
satisfying the inequalities a — Xq < Xi < • • ■ < xn- \ < xn — b is called a
partition o f \a,b]. The interval [xk- i , x k], k = 1,2, . . . , n is called the kth
subinterval o f P and we write Axk = Xk — %k-1> so that
n
Azfc = b — a.k = 1
The collection o f all possible partitions of [a,b ] is denoted by p[a, b}.
Definition 1.1.4. Let f be a real-valued function defined on [a, b]. I f P —
{ xq, x i , . . . , x n} is a partition o f [a,b], we set A f k = f ( x k ) — f { x k - 1)> k =
1 ,2 , . . . , n. I f there exists a positive number M such that
E ia /*i ^ Mk = l
fo r all partitions o f [a, b], then f is said to be o f bounded variation on [a, 6].
We now give some theorems that help us to decide when a function is of
bounded variation.
Theorem 1.1.1. Let f be a real-valued function defined on [a, 6]. I f f is
monotonic on [a, b], then f is o f bounded variation on [a, b].
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CHAPTER 1. JOHN’S THEOREM 7
Proof. As / is monotonic on [a, b], by D efin ition 1.1.2 / is either increasing
or decreasing on [a, b]. Suppose / increasing on [a, b\. Then, fo r any pa rtition
{x 0, x i , . . . , xn} o f [a, 6], we have f { x k) - i) > 0 giving A f k > 0. Now
53 ia ^ i = i t Afk = i t u ^ ) ~ = ~k = l k = 1 fe=l
so Y^k=i |A/fc| < M w ith M ~ f (b) — f (a) . Thus / is of bounded variation
on [a, 6]. I f / is decreasing on [a, 6] then —/ is increasing on [a, b] and the
above argument again shows th a t / is of bounded variation on [a, 6]. □
T heorem 1.1.2. Let f be a real-valued function defined on [a, b]. I f f is con
tinuous on [a, b] and i f f exists and is bounded in the interior, say \ f ( x ) \ < A
fo r all x in (a, b), then f is o f bounded variation on [a, 6],
Proof. Let { x q , x x, . . . , xn} be a p a rtition o f [a, 6], Applying the mean value
theorem to the subinterval [xk- X, xk], we have
A /* = f ( x k) - f ( x k- 1) = f ( t k) (xk - x k- x) = f ( t k) A x k
for some t k G (xk- X, x k). Then
n n n n
53 = 5 3 i/W A z fc i = 53 i/ '(4 ) iA£ft < ^ 5 3 a ®*.k—1 fc= l k = l k = l
Hence X X = i |A/fc| < M w ith M = A(b — a) so f is o f bounded variation on
[a, b]. □
The function f Ix2 cos—, x ± 0,
0, x = 0
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CHAPTER 1. JOHN’S THEOREM 8
is of bounded variation on [0,1] as / is continuous on [0,1] and
f sin — + 2x cos —, x ^ 0,m = \ x
^ 0, x = 0,
and
I/ '( z ) l < 3.
We note that f need not be bounded for / to be of bounded variation.
For example, f ( x ) = x 1/3 is monotonic so / is of bounded variation in every
finite interval (Theorem 1.1.1). However, f ' { x ) = ^73 —» +00 as x —» 0.
Not all continuous functions are of bounded variation. Consider the func
tion
{ a: sin ( —) , 0 < x < 2,\ x ) ’
0, x = 0,
which is continuous on [0,2], and the partition2 , U , 2
2n — S ’ 5 ’ 3
Here
*o = 0, x * = 2 n - ( 2 k - l ) ,k = 1,2, ' " , n ‘
For k = 2 , . . . , n we have,
A /* = f ( x k) - f ( x k- i )2 . ( (2n — (2k — l) ) 7r
sin 1 "2n — (2k — 1) \ 2
2 • ( (2n - (2k - 3) )n\2 n - ( 2 k - 3 ) V 2 /
9 9________ / 1 'jn —fe __________“ ______' / 0 7- 1 o v /2n — 2k -f-1 271 — 2k 3
/ 2 2( - 1) o ?T7- r - r +2n — 2k + 1 2n — 2& + 3
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CHAPTER 1. JOHN’S THEOREM 9
Hence
n n
E l A A l > e i a mk = l k = 2
2n - 2fc + 1 2n — 2k + 32 2
/ is not of bounded variation on [0, 2] as lim — = +oo.n—»+oo / K
T heo rem 1.1.3. Let f be a real-valued function on [a, 6]. I f f is o f bounded
variation on [a, 6], say X)|A/fc| < M fo r a ll partitions o f [a,b], then f is
bounded on [a,b]. In fact, \ f (x) \ < |/(a )| + M fo r x 6 [a, 6].
Proof. Let x € (a, b) and consider the p a rtition P = {a,x,b} . As / is of
bounded variation on [a, 6], we have
l / ( * ) l = I / O * ) “ / ( a ) + / ( a ) I < l / ( * ) “ / ( a ) I + l / ( a ) l < M + | / ( a ) |
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|/ ( * ) - /(a ) | + | / ( 6) - / ( * ) | < M,
which gives | f ( x ) — f (a ) | < M . Now
as required.
CHAPTER 1. JOHN’S THEOREM 10
If x = a, the inequality holds trivially. If x = b, the partition {a, b} gives
]/(&)| < 1/(6) - f ( a ) I + |/(o )| < M + \f(a)\. Hence |/(x )| < |/(a )| + M for
all x G [a, b}. □
The converse of Theorem 1.1.3 is not true. The function
{ 7Txs i n—, 0 < x < 2,
*0, x = 0,
is bounded on [0,2] as |/(x )| < |x| < 2 but is not of bounded variation on
[0,2].
Definition 1.1.5. Let f be a real-valued, function defined on [a,b]. Let f
be o f bounded variation on [a,b], and let J^(P ) denote the sum Y^k= i |A/fc|
corresponding to the partition P = {xo, x i , . . . , xn} o f [a, 6]. The number
V, = V /(a, b) = sup { Y ^ ( P )|P 6 p m }
is called the total variation o f f on the interval [a, 6].
Theorem 1.1.4. Let f and g be real-valued functions defined on [a,b]. Sup
pose fu rther that f and g are both o f bounded variation on [a, b]. Then so are
their sum, difference and product. Also, we have
vf±g < V f + Vg and Vfg < AVf + BVg,
where A = sup{|#(x)| |x G [a ,6]} and B = sup{]/(x)| |x G [a, 6]} .
Proof. We just prove the second of these two inequalities. Consider h(x) —
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CHAPTER 1. JOHN’S THEOREM 11
f (x)g(x) . Then for each p a rtition P o f [a, 6], we have
\Ahk\ = \ f ( xk)g(xk) - f ( x k - i ) g ( x k- i ) \
= If { x k)g{xk) - f { x k- x)g{xk)\
+ \ f ( x k- i ) g ( x k) - f ( x k- i )g(xk- i ) \
< A \A fk\ + B\Agk\.
Hence h = f g is o f bounded variation and V/g < AVf + BVa. □
There is a close relation between monotonic functions and functions of
bounded varition. We have already seen th a t monotonic functions are always
of bounded variation (Theorem 1.1.1), and we w ill see shortly th a t functions
of bounded variation can always be w ritten in terms o f monotonic functions
(Theorem 1.1.8). Being o f bounded variation is a stronger condition than
m onotonicity, indeed, the sum or product of monotonic functions need not
be monotonic. For example, x and —x2 are monotonic on [0,1], but x — x2
is not; and x is monotonic on [—1, 1] but x2 is not.
I t should be noted th a t the reciprocal of a function of bounded variation
is not necessarily of bounded variation. For instance, suppose f ( x ) —> 0 as
x —» c. Then 1 / / is not bounded in any interval containing c. Hence by
Theorem 1.1.3, 1 / / is not o f bounded variation on such an interval. Ex
cluding functions whose values get a rb itra rily close to zero le t us extend
Theorem 1.1.4 to quotients.
Theorem 1.1.5. Let f be o f bounded variation on [a,b] and assume that f
is bounded away from zero, that is there exists a positive number m such that
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CHAPTER 1. JOHN’S THEOREM 12
0 < m < \ f ( x ) \ fo r all x in [a, 6]. Then g = I f f is also o f bounded variation
on [a, b], and Vg < ^
Proof. We have
|A 0fc| =1 1 f ( x k- 1) - f { x k)) A f k
f(Xk) /(Zfc-l) f { x k) f ( x k- 1) f (X k ) f (x k- 1)< jA /fcl
m 2
□
Theorem 1.1.6. Let f be a real-valued function defined on [a, b]. Let f be o f
bounded variation on [a, 6], and assume that c € (a, 6). Then f is o f bounded
variation on [a, c] and on [c, b] and we have
Vf (a,b) = Vf (a,c) + Vf (c,b).
Proof. Consider partitions P i of [a,c] and P2 o f [c,b]. Then P0 = P i U P2 is
a pa rtition o f [a, b\. Denote by ]P (P ) the sum |A ./*| corresponding to a
p a rtition P . Then
J ^ iP l ) + = £ (i> „ ) < V,(a,b)-
Hence X X P i) and XX-^2) are bounded by Vf(a, b) so / is of bounded variation
on [a, c] and [c,b\. Now
£ ( P 0 + X ; « ! ) < V f M ) = * V,{a,c) + Vf ( c , b ) < V f (a,b).
For the reverse inequality, consider a pa rtition P = {xq,Xi , . . . , xn} <E
p[a, b] and Po = P U {c } the p a rtition obtained by adjoining the point c. I f
c 6 [£fc_i, xk\ then
l/(zfe) ~ f ( x k - 1)| < | f ( x k) - /(c)| + |/(c) - / ( z fe_i)|
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CHAPTER 1. JOHN’S THEOREM 13
giving X )(P ) ^ E M - The points of P0 in [a, c] give a p a rtitio n P i of [a, c]
and the points in [c, 6] give a pa rtition P2 o f [c, 6]. Now
E(p) s £«) = E(p>)+Ewo c)+*)•This shows th a t Vf(a, c) + Vf(c,b) is an upper bound for every sum ]T)(P).
Since th is cannot be smaller than the least upper bound, we have
V f(a ,b )< V f (a,c) + Vf (c,b).
The asserted equality now follows from the two inequalities. □
Theorem 1.1.7. Let f be a real-valued, function defined on [a, b] and let
f be o f bounded variation on [a, b]. Let V be defined on [a, b] as follows:
V (x) = Vf(a, x) i f a < x < b , V(a) = 0. Then:
(i) V is an increasing function on [a,b].
(ii) V — f is an increasing function on [a,b].
Proof. For a < x < -y < b, w rite Vf (a, y) = Vf (a, x) + Vf(x, y). Then
V(y) - V(x) = Vf (a,y) - Vf (a,x) = Vf (x,y) > 0.
Hence (i) holds.
Consider D (x) = V(x) — f { x ) i f x £ [a, 6]. For a < x < y < b ,
D (y ) - D ( x ) = V(y) - V ( x ) ~ [ f (y) - f ( x ) ]
= Vf (x,y) - [ f {y) - / ( * ) ] .
By the definition of Vf(x, y), we h a ve / (y)—f { x ) < Vf(x,y) so D ( y ) —D(x) >
0 and (ii) holds. □
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CHAPTER 1. JOHN’S THEOREM 14
T h e o rem 1.1.8. Let f be a real-valued function defined on [a, 6]. Then
f can be expressed as the differencef is o f bounded variation on [a, b] 4=>
o f two increasing junctions.
Proof. (= > ) Suppose / is of bounded variation on [a, 6]. W rite f = V — D
where D is the function from the proof of Theorem 1.1.7. Now V and D —
V — f are increasing by Theorem 1.1.7.
(4 = ) Suppose f = g — h where g, h are increasing. Then g, h are mono
tonic and so are o f bounded variation by Theorem 1.1.1. Now g — h is o f
bounded variation by Theorem 1.1.4, and so / is of bounded variation. □
This representation is not unique! I f / = f i — fa is one representation of
/ as the difference o f two increasing functions, then f = ( f i +g ) — (/2 + g),
where g is an a rb itra ry increasing function, is another representation o f / .
I f g is strictly increasing, so are f i + g and / 2 + g, therefore Theorem 1.1.8
holds if “ increasing” is replaced by “s tric tly increasing” .
T heo rem 1.1.9. Let f be a real-valued function defined on [a,b] and let f
be of bounded variation on [a, 6]. I f x G (a, 6], let V (x ) = Vf(a,x) and put
V(a) = 0. Then every point o f continuity o f f is also a point o f continuity
o f V . The converse also holds.
Proof. (4 = ) V is monotonic as i t is increasing, hence the righ t- and left-hand
lim its V ( x + ) and V ( x - ) exist fo r each point x in (a, b). By Theorem 1.1.8,
the same is true of f ( x + ) and f ( x - ) .
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CHAPTER 1. JOHN’S THEOREM 15
I f a < x < y < b, then 0 < |f ( y ) — f ( x ) \ < V( y ) — V(x) by the definition
o f Vf(x,y) . Letting y - * x, we see 0 < |f ( x + ) - f ( x ) \ < V{x-\-) — V(x) .
S im ilarly, 0 < | f ( x ) — f ( x —) \ < V(x) — V ( x —). These inequalities im ply
th a t a po in t o f continu ity of V is also a point o f continu ity of / .
(= ^ ) Let / be continuous at the point c in (a, b). Given e > 0, there
exists S > 0 such that 0 < \x — c\ < 5 implies 0 < \ f (x) — f (c) \ < e /2. For the
same e, there exists a partition P of [c, 6], say P = {xo, x i , . . . , xn}, x0 = c,
xn — b such that
fc=l
Adding more points to the pa rtition can only increase the sum Yh |A/fc|> so
we may assume 0 < x\ — xq < 8. This means tha t
IA/i| = | /(xi) - / ( c ) | <
As { x i ,X 2 , . . . , xn} is a p a rtition of [xi,b\, we have
V>(c,») - | < | + E lA A I < | + V f a , b).k=2
Hence
Vf (c,b) - V f {xu b) < e.
B ut 0 < V{x \ ) — V(c) = Vf ( a ,x i ) - Vf(a,c) = Vf (c ,x i) = Vf(c,b) —
Vf(x i ,b ) < e. Hence 0 < x x — c < 6 implies 0 < V '(x i) — V(c) < e. This
shows V (c+ ) = V(c). S im ilarly, V (c—) — V(c). □
T heorem 1.1.10. Let f be continuous on [a, 6]. Then
f is o f bounded variation / can be expressed as the difference
on [a, b] o f two increasing continuous Junctions.
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CHAPTER 1. JOHN’S THEOREM 16
Proof. W rite / = V — D as in Theorem 1.1.7. Then V and D are increasing
by Theorem 1.1.7, and are continuous by Theorem 1.1.9. □
As in Theorem 1.1.8, Theorem 1.1.10 holds i f “ increasing” is replaced by
“s tric tly increasing” .
Theorem 1.1.11.
/ is o f bounded variation f b= > the Riemann integral / f ( x ) dx exists,
on [a, b] Ja
Theorem 1.1.11 is proved in [4, pp. 207-212].
The converse of Theorem 1.1.11 does not hold, th a t is, a function may be
Riemann integrable w ithout being o f bounded variation. Consider
/ ( * ) = ( ° < :C- 1’[ 0, x = 0.
Then / is defined on [0,1]. As
J r f ( x ) dx = £ dx = [2x ll2}\ = 2 - 2e1/2,
we have f { x ) dx = 2 so the Riemann integral exists. Consider the pa rti
tion P o f [0, 1] w ith P = { 0, J , J, J , . . . , a } . Then A / i = f ( x i ) - f ( x Q) = n i
Suppose f is o f bounded variation on [0,1]. Then there exists a positive num
ber M such th a tn
n 1/2 = IAAI < I ^ M (Vn)>Jt=i
but th is fails whenever n > M 2. Hence / is not of bounded variation on
[0, 1].
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CHAPTER 1. JOHN’S THEOREM 17
1.2 John’s theorem
F irst we give John’s theorem, which was stated and proved by John in 1934
in [20],
T heo rem 1.2.1. Let f be a real-valued function on (—oo,+oo) which is
periodic o f period 1 and of bounded variation on [0,1]. I f c > 1 is a given
rational number, then
£ ^ f ( S ) = lo g c i m i v ' (1 2 1 )
where an(c) is defined as follows: set c = p/q, where p and q are coprime
integers with q > 0, then
an(c) = <
0, i f P in , q fn ,
-p , i f p \ n , q fn ,
q, i f p f n , q \ n ,
q - p , i f p \ n , q \ n.
Our form ulation is sligh tly different from John’s original statement which
is as follows:
T heo rem 1.2.2. Let g be a real-valued function on (—oo, +oo) which is
periodic o f period 1 and of bounded variation on [0,1]. I f c > 1 is a given
rational number, then
£ { * ■ - i | i ) = logci 9{y) dy ' (1-2-3>fo r the same definition o /a n(c) and fo r any t G R.
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CHAPTER 1. JOHN’S THEOREM 18
We show th a t the two versions of the theorem are equivalent.
Proof. (Theorem 1.2.1 = > Theorem 1.2.2) Let g be a real-valued function
on (—oo, +oo) which is periodic of period 1 and of bounded variation on
[0,1]. Let c > 1 be a given rational number, and let £ G l .
Set f ( x ) = g(t — x ). Then / is a real-valued function of (—oo, oo). Note
th a t / is periodic of period 1 as f ( x + 1) = g(t — (x + 1)) = g(t — x — 1) =
g ( t - x ) = f (x ) .
Let {xo = 0, x i , . . . , x n = 1} be a pa rtition of [0,1].
I f t e Z we set = 1—xn_fc, k = 0 ,1 , . . . , n, so th a t {xq = 0,x'1, . . . , x ,n =
1} is a pa rtition of [0,1]. Since g is of bounded variation on [0 ,1], there exists
M > 0 such th a t
~ 9 {x 'k- 1 ) | < M ./c = l
Hencen
^ | p ( l - xn—k) - t f ( l - x n_fc+ 1 ) | < M .fc=i
As g is periodic w ith period 1 and t G Z , we have71
5 3 19 {t - xn- k) - g(t - xn_fc+i)| < M.k= 1
Next, as f ( x ) = g(t — x), we deduce tha t
n
^ 2 \ f ( Xn-k) ~ f(Xn-k+l) I < M, k=l
th a t isn
5 3 l / f a n —fc + l ) “ f ( x n - k ) I < M. k= 1
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CHAPTER 1. JOHN’S THEOREM 19
Changing the summation variable from k to I = n — fc + l , we obtainn
1=1
so tha t / is of bounded variation on [0, 1].
I f t ^ Z, then
Xq = 0 < t - [t] < 1 = xn.
Let m be the unique integer € {0 ,1 , . . . , n — 1} such tha t
t M ^ 3'm+l-
Set
x'Q = 0,
x [ = t - [i] + 1 - xn,
®n—m t — [t] + 1 — X m+ i ,
* n—m+1
Then {x'0, x'x, . . . , x'n_mJrl} is a pa rtition of [0,1]. Since g is o f bounded vari
ation on [0, 1], there exists M \ > 0 such th a t
n—m + l
X ) Is K ) -0 (a 4 - i) l ^ M i-k= i
Hencen—m
1 “ M + 1 “ x n + l - k ) - g ( t - [t] + 1 - Zn+2-fc)|k=2
n—m
fc=2
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CHAPTER 1. JOHN’S THEOREM 20
As f ( x ) = g(t — x ) we deduce tha t
n —m
53 l / ( M - 1 + Z n + l- fc ) - f ( [ t ] - 1 + X n +2-fc )| < M i .
k=2
As / is periodic of period 1, we obtain
n —m
53 l / ( X n + l- i fc ) ~ / ( z « + 2 - f c ) l < M i .
k= 2
Changing the summation variable from k to I = n — k + 1 , we obtain
n —1
53 \ f ( x i ) - f ( x i + i ) \ < M i,l=m+1
tha t is,
equivalently
n —1
53 i/c^+i) - /(^)i < Mi,l=m+1
5 3 — /(^ fe - i) i < M i.k = m + 2
As <? is of bounded variation on [0,1], by Theorem 1.1.3 g is bounded on [0,1].
As g is periodic o f period 1, g is bounded on (—oo, oo). As f ( x ) = g(t — x),
f is bounded on (—oo, oo), say |/(x ) | < K for a ll i £ l Hence
n n
\ f ( x k ) ~ f ( X k - l ) \ = \ f ( x m+1) ~ f ( x m) \ + \ f ( x k) - f ( x k- i ) \k=m + 1 k = m + 2
< | / ( Z m + l ) | + l / ( * m ) | + M i
< 2 K + M i.
S im ilarly by form ing a pa rtition from Xq, Xi , . . . , xm we can bound J2k=i I f ( x k)-
f ( x k — 1)|, proving th a t / is o f bounded variation on [0, 1].
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CHAPTER 1. JOHN’S THEOREM 21
Lastly using Theorem 1.2.1, we have
y an(c) / logn \ _ y an(c) , f \ogn^ n 9 \ lo g c ) ^ n V logC>
: f f ( y )Jo
logo I f ( y ) dy )
log c g(t — y) dy Jo
r t - lr t - l= —logc g(z) dz (w ith z = t — y)
= log c g(z) dz Jt-1
= log c g(y) dy.Jo
(Theorem 1.2.2 =4> Theorem 1.2.1) Let / be a real valued function on
(—00, 00) which is periodic o f period 1 and of bounded variation on [0, 1].
Let c > 1 be a given rational number.
Set g(x) — f ( —x). Then g is a real valued function on (—00, 00). Note
tha t g is periodic of period 1 as g(x + 1) = f ( —(x + 1)) = f ( —x — 1) =
f ( - x ) = g(x).
Let xo = 0, x i , . . . , xn = 1 be a p a rtition o f [0,1]. Then
x'Q = 1 - Xn, x'x = l — rrn_ i , . . . , x'n = 1 - x0
is also a pa rtition o f [0, 1].
Since / is of bounded variation on [0,1], there exists M > 0 such th a t
X ) If ( x 'k) - / (Zfc-i) l < M and J 2 1/(1 - x'n_k) - / ( I - < _ fc+1)| < M.fc = l k = l
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CHAPTER 1. JOHN’S THEOREM 22
As / ( I - x'n_k) = f ( - x n- k) = g(xn- k), we have
n
23 \9(Xn-k) ~ g(Xn-k+1)| < M.fc=1
Setting j = n — k + 1 gives 2 3 \g(xj~i) - g(xj)\ < M , and, as we mayj = l
change the order due to the absolute value,
71
- 9 i xj - i ) \ < M >3=1
so g is o f bounded variation on [0, 1].
Lastly, setting t — 0 in Theorem 1.2.2, we have
an{c) f l o g n \ _ ^ an(c) f log n \E u n W £ I iV & n \ _ V - a n \ u) t , _
n 7 Vlog c ) ~ ^ n 9 \71=1 \ o / n _ j \
= log c [ g(y) dyJo
= log c I f ( —y) dy I of f i - v )Jo
= log c f ( l - y ) d y Jo
= —logc f ( z ) dz (w ith z = l — y)Jo
= log c [ f ( z ) dz,Jo
as required. □
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CHAPTER 1. JOHN’S THEOREM
1.3 Proof of John’s theorem
23
Let c > 1 be a rational number. Set c = p/q, where p G Z , q € N and
gcdfp, q) = 1. Set A = — . Let a e E b e such th a t 0 < a < 1. Let M e N.logc
We begin by lis ting a few simple properties o f c,p, q, A, a and M th a t we
w ill need in the proof of John’s theorem.
Lem m a 1.3.1. (i) p > q,
(ii) log c > 0,
( iii) A, A-1 > 0,
(iv) a + A log M q < A log Mp,
(v) {A lo g M p } = {A log M g },
(vi) . logq = lQgP 1logP — log? log p log q
Proof, (i) As c = p/q > 1 and q > 0 we have p > q.
(ii) As c > 1 we have log c > log 1 = 0.
(iii) By (ii) we have
1 A 1 1> 0, — = log c > 0.logc A
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CHAPTER 1. JOHN’S THEOREM 24
(iv) We have
a < 1 = » a < AA-1 =>• a < A log c
=$> a: < A log - ==> a < A log” & q ~ ° M q
==> a < A (log M p — log Mq)
= » a + A log M q < A log Mp.
(v) Also
{A log M p } — { A log M p — A log M q + A log M q }
= {A log ^ + A log M q }
= {A • A-1 + A log M q ]
= {1 + A log M g}
= {A log M g }.
(vi) F ina lly
logg — (log p — log g) + log p logp= —1 +
logp — log q logp — log q log p — log q
Next we recall Euler’s constant
7 = lim -j — log = 0.5772156649__
We w ill need the follow ing estimate.
Lem m a 1.3.2. Let x ,y € R be such that x > y > 0. Then
£ I = l o g ( * ) + < > ( ! ) .£ £ m \ y j \ y j
y < m < x
as y +oo, where the constant implied by the O-symbol is absolute.
□
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CHAPTER 1. JOHN’S THEOREM 25
Proof. The asym ptotic formula
T - = logy + 7 + O ( - ) , y -> oo ^ m V y Ime N m < y
is well-known, see for example [3, p. 55]. The im plied constant is absolute.
Thus for x > y > 0 we have
1E 1 = E 1-Em " , to " , tom€N meN meN
y < m < x m < x y < m
= ^ lo g x + 7 + 0 - ^ logy + 7 + O Q
= iosG)+oS)’as y —» oo. □
We now make use of Lemmas 1.3.1 and 1.3.2 to prove the follow ing result.
Lem m a 1.3.3.
lim Y — = ctrA-1 .M -* o o TOmeN
M q < m < M p 0 < {—A log m }<a
Proof. We have
E 1 = E i" , TO " , TOm€N m€N
M q < m < M p M q < m < M p0 < {—A logm }<a 0 < —A log m —[—A log m ]<a
E -1TO^ez meNE
A log M q < £ < \ log JVf p - f l M q < m < M p0 < —A lo g m —[—A lo g m ]< a
[— A lo g m ] = —I
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CHAPTER 1. JOHN’S THEOREM 26
Elez
A log M q < t< —\ log M p+1
E€eN
A log Mq<£<A log M p+1
E£ez
A log Mq<£< A log M p+1
E 1m£N
M q<m <M p 0 < —A log m + i< a
[—Alogm]= —£
E -1me N
M q<m <M p 0 < —A log m+£<a
me N M q<m <M p e-a t
e A < m < e *
We now break the sum over £ into 4 subsums S i, S3, S4 according to the£ _ Q ^
sizes o f e“ and e* relative to M p and Mq. We have
E E - = Si+S2 + S3 + S4,£eZ meN
A log M q < t< \ log M p+1 M q<m <M pI —C*
where
e a <m<e>
= E E s-iez meNA log M q < £< \ log M p+1 M q<m <M p
e-a e a < M g
M p <e^
= E E s-£ez meN
A log M q < £< \ log M p+1 fcr.aI — Of
Mq<e A
Mp<e^l—Ot
e A < M p
* - E E s-£ez meN
A log M q <£< \ log M p+1 M ?<rn< efe~TT<Mq
e% <M p
e A < m < M p
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CHAPTER 1. JOHN’S THEOREM 27
s<= E £ b£ez me nA log M q <£< \ log M p +1 1
£ q C A <771^6 AMq<e A
< M p
We note th a t S2 = 0 unless e1 2 < M p. Also in the sum S3 we observe tha t
M g < e* as A log M g < A
F irst we examine S i. By Lemma 1.3.2 we have
* - £ M 9 * ° ( i ) )A log M q<£<A log M p+1 A log M p<£<a+A log Mq
= 0,
as a + A log M q < A log M p by Lemma 1.3.1 (iv).
Next we determine S2. By Lemma 1.3.2 we have
* = E M ¥ s ) + ° ( - k£€Z
A log M q < £< \ log M p+1 c*+A log Mq<£<a+A log Mp
A log Mp<£
, _ , <-ae a / \ e a .
- a J£eZ ' \ o A /
A log Mp<£< A log M p+as M ; ? M +
by Lemma 1.3.1 (iv). We consider 3 cases according as {A log M p } = 0,
0 < {A log M p } < 1 — a, o r l — a < {A log M p } < 1.
I f {A log M p } = 0 then A log M p G Z so i = A log M p. Thus
& = log + O ^ 1g(A log M p —a )/A J \ g(A log M p —a)/A
- + o ( — \. a v m :
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CHAPTER 1. JOHN’S THEOREM 28
I f 0 < {A log M p } < 1—a then [A log M p \+ 1 > A log M p + a so A log M p £
Z and no such integer £ exists so S2 = 0.
I f 1 — a < {A log M p } < 1 then A log M p Z and [A log Mp] + 1 <
A log M p + a and £ = [A log Mp] + 1. Then
M p \ ^ / 1S2 log
log
g([A log M p]+1—a)/A
M p
+ 0
g(A log M p —{A log M p }+ 1 —a)/A
1+1A log Mp) ^ / 1 \
^ M J
e ([A log M p]+1—a)/A
+ 0 g(Alog Mp— {A log M p }+ l-c t) /A
Hence
log (e a " a“ J + 0
& 1 + {A log M p } ( J _ 'A + V M .
i +0(s)-S2 = 0,
a - l + {A lo g M p } ( 1 ' A + ° [ m .
{A log M p } = 0,
0 < {A lo g M p } < l —a;,
1 — a < {A log M p } ,
Now we tu rn to the evaluation of S3. We have by Lemma 1.3.2
1S3 = E
tezA log Mq<£<A log M p+1
£<o+Alog M q £ < \ log M p
h ^ W q ) + 0 K M
Eeez
A log M q<£<a+A log Mqlog|^ ) +0( s /
by Lemma 1.3.1 (iv).
Here we consider 3 cases according as {A log M q } = 0, 0 < {A log M q } <
1 — a, or 1 — a < {A log M q } < 1.
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CHAPTER 1. JOHN’S THEOREM 29
I f {A log M q } = 0 then A log M q € Z and we have
I f 0 < {A log M q } < 1 - a then [A log Mq] + 1 > a + A log M q so
A log M q 0 Z and no such integer £ exists so S3 = 0.
I f 1 — a < {A log M q } < 1 then [A log M g] + 1 < A log M q + a and we
have
S3 = log[A log Mql-fl \
e a
M q
[A log Mq] + 1H i
log M q + O l —
1 — {A lo g M g } + 0 / J _ AA M .
Hence
Ss =
0 { i ) ,
0,1 — {A log M q }
A
which is equivalent to
o ( A ) ,s3 = < 0 ,
1 — {A log M q }
{A log M g } = 0,
0 < {A log M g } < 1 — a,
+ 1 - » < {A lo g M g },
{A log M p } = 0,
0 < (A log M p } < 1 — a,
+ ° [ j j r ) > 1 - a < {A lo g M p },
as (A log M p } = {A log M g }, by Lemma 1.3.1 (v).
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CHAPTER 1. JOHN’S THEOREM 30
Lastly we consider S4 . By Lemma 1.3.2 we have
A log M q<£<A log Mp+1 a+A log Mq<£< A log Mp
a+A log Mq<£<A log Mp
We consider 3 cases according as {A log M p } = 0, 0 < {A log M p } < 1—a,
or 1 — a < {A log M p } < 1.
I f {A log M p } = 0 then A log M p e Z and no such integer I exists so
SA = 0.
I f 0 < {A log M p } < 1 — a then I = [A log Mp] and
I f 1 — a < {A log M p } < 1 then no such integer £ exists and S± = 0.
Hence
P utting i t a ll together, we now look at the required sum S\ + S2 + S3 + 64
in 3 cases according as {A log M p } = 0, 0 < { A log M p } < 1 — a, o r l — a <
{A log M p } < 1.
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0 {A log M p } — 0,
0 < { A log M p } < 1 — a,
1 — a < {A log M p } < 1.
CHAPTER 1. JOHN’S THEOREM
I f {A log M p } — 0 then
S l = 0 , f t = 2 + 0 ( i ) . S 3 = o ( i ) , S 4 = 0 ,
so
S± + S2 + £3 + S4 = — + 0 •
I f 0 < { A log M p } < 1 — a then
Si = 0, S2 = 0, £3 = 0, SA = j + o ( j j ) ,
so
Si + Si + Ss + S ^ j + O ^ y
I f 1 — a < { A log M p } < 1 then
S1 = 0, ga = a ~ 1 + ( l0gM p}+ ° Q _ ) .
S4 = 0, g3 = l - { A b g M p } + 0 Q _ ^
SO
Sl + S , + 8, + 3t = ^ - l + {AI°gMp} + l - { A l o g M r f + 0 / lA V 1VI
= x+0(f )-Hence in a ll three cases we have
s1 + s2 + s3 + s4 = j + o (J ^ J
so
lim V ' — =M-* 00 <' m AmCN
Mq<m<Mp 0<{—Alog m}<a
as required.
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CHAPTER1. JOHN’S THEOREM 32
For a € [0,1] we define a special step function <j)a as follows: for y G R
we set
M y ) = 0) y e R,
0 i ( i ) = 1) y € M,
and for a ^ 0,1
M v) = { x ’ 0 “ W < (1-3.1)[ o, a < { y } < 1.
Lemma 1.3.4.
lim V "' — (f>a(—A logm ) = aA-1 .M -* oo 771
m€N M q<m<M p
Proof. As </>o(y) = 0 for a ll y, Lemma 1.3.4 is tr iv ia lly true for a = 0 and we
may assume th a t 0 < a < 1.
By the defin ition o f <pa we have
£ ^ f e ( - A l ° g m ) = £ 1m 6N meN
M q<m<M p M q<m<M p0 < { — A logm }< a
and the asserted result follows by applying Lemma 1.3.3. □
Lemma 1.3.5. Let f be a periodic step function o f period 1. Then
d im — / ( - A l o g m ) = A-1 / f ( y )d y .
M q<m <M p
Proof. As / is a periodic step function of period 1, / can be b u ilt up as a
linear combination o f a fin ite number of the step functions 4>a(y) w ith differ
ent parameters a. Hence the asserted result holds by repeated application of
Lemma 1.3.4. □
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CHAPTER 1. JOHN’S THEOREM 33
Lemma 1.3.6. Let f be a periodic function of period 1, which is Riemann
integrable on [0,1], Then
lim — /( -A lo g m ) = A-1 f f ( y ) dy.M~°° m J 0
M q<m <M p
Proof. Let e > 0. As f ( y ) is a Riemann integrable function on [0,1] and is
periodic of period 1, there exist two step-functions and <£(?/) o f period
1 such tha t
m < f ( v ) < H v ) (1-3.2)
and
[ (<% ) - 4>{v)) dy < e, (1.3.3)Jo
see for example [5, Sections 1.1, 1.2, pp. 11-26]. Set
Sm {4>) = 53 - 0 ( -A log to), to
me N M q<m <M p
Sm ($) = 53 —$(~A logm).mmeNM q<m <M p
Then, by Lemma 1.3.5, we have
lim SM{<f>) = A-1 [ 4>{y) dy,M —> oo J Q
lim SM($) = A-1 [ $(y) dy, (1.3.4)M —*oo J o
and from (1.3.2) we have
Sm {4>) < 53 - / ( - A l o g m ) <m me N M q<m <M p
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CHAPTER 1. JOHN’S THEOREM 34
Prom th is and (1.3.4) we conclude
/*i______________________ i1 4>{y) dy < lim in f V " — / ( - A l o g m )
J o Z a mM q<m <M p
< l im sup ^ 2m€N m
M q<m <M pM q<m <M p
JO
and applying (1.3.3) gives the asserted result. □
We now have the results needed to prove John’s theorem (Theorem 1.2.2).
Proof. We follow the proof of John’s theorem given by Rademacher in [27,
pp. 170-173], but expanding on the details where necessary. We note th a t
the proof only requires g to be Riemann integrable (recall from Theorem
1.1.11 th a t functions of bounded variation are always Riemann integrable).
We wish to study the expression
for a Riemann integrable function g and an(c) as defined in (1.2.2). We begin
by showing tha t we need only consider such N which are divisib le by pq.
For 0 < R < pq, as an(c) and g are bounded, we have
i™ E ir9(i_Alogn)n = 1
Npq+R Npq+R
g(t — X log n) < A ^n=Npq+ln=Npq+ 1
fo r some constant A.
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CHAPTER 1. JOHN’S THEOREM
By Lemma 1.3.2 we have
Npq+R 1
n=Npq+ 1
= log
1 nNpq<n<Npq+R
Npq + R N+0(vNpq
= log{ I + + 0 ( j j
Hence
proving our claim.
Now set
N p q )
lim Y — = log 1 = 0,oo *n.
Npq+R
N -* oo e~~J n n=Npq+ 1
Mpq / \
-5m (5) = E “ A logn).1n = l
Then, using the definition of an(c), we obtain
Sm (9) = E A l o g n ) - E - 0( t - A l o g n )* * r?. •> ■ * r i
l<n<M pqq|n
Mp
\ < n < M p qp|n
Mq" y 2 2= y \ - g ( t - A logm ?) - V - <?(* - A logm p).
i TO ' TOm = l m = l
Using Lemma 1.3.1 (v i), we have
log to + log qA log mq =
which gives
? (i - A logm ?) = g ( t ■
as g is periodic o f period one. As
g ( t - A logm p) = ? ( t -
log TO+
logp - 1,logp — log q logp — log q logp — log q
log to logplog P - log q logp - log q
log to logp \logp - lo g q logp log q ) ’
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CHAPTER 1. JOHN’S THEOREM 36
we deduce tha t
log p log q log p log qlog m log p
We set t = A logp, for i f (1.2.3) is true for any special value to, i t is also
1.4 Evaluation of certain infinite series using
John’s theorem
We begin by choosing f ( x ) = 1 (x £ R). C learly / satisfies the conditions of
John’s theorem.
T heo rem 1.4.1. Let p and q be coprime integers w ithp > q > 0. Then
true for any other t, as g{x) and g(x — to + t) regarded as functions of x are
both periodic and Riemann integrable.
Hence a ll we need to show is
Mp .
lim SM(g) = lim V — s (-A lo g m ) = A' 1 / g(y) dy
B ut th is was shown in Lemma 1.3.6, and the proof is complete. □
where an is defined in (1.2.2).
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CHAPTER1. JOHN’S THEOREM 37
Taking p = 2 and q = 1, we obtain the well known result
1 - - + = log 2.2 3 4 &
W ith p = 3 and q = 1 we have
, 1 2 1 1 2 1 1 2 , 01 H------------ 1------1------------ j------1------------ 1------ = log3.
2 3 4 5 6 7 8 9 6
W ith p = 3 and q = 2 we obtain
2 3 2 1 2 3 2 1 , 3---------- 1------------ 1------------ 1----------------(- .. . = Jog —.2 3 4 6 8 9 1 0 12 6 2
These series are a ll o f the form
00E Qn
n ’7 1= 1
where {a n} is a repeating sequence of integers. Such a series is called a
harmonic-type series. The harmonic series itse lf demonstrates tha t not a ll
harmonic-type series converge. Lesko [23] has recently established a necessary
and sufficient condition for a harmonic-type series to converge, namely
oo
A harmonic-type series — with repeating coefficients a i,a 2 , . . . , a ki n
n = lk
converges i f and only i f ^ a* = 0.i = 1
Series o f the formoo
E O-n k r - ’
n = l
where { a „ } is a repeating sequence o f integers and A: is a given real number,
have been treated by Longuet-Higgins [25].
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CHAPTER 1. JOHN’S THEOREM 38
For an integer b > 1 i t is convenient to define
f b - 1, i f b | n, en = €n( b ) = { (1.4.1)
[ - 1, i f bj(n,
as in [8, p. 22] and [12, p. 261]. C learly
an(b) = -e n(&), en(2) = ( -1 ) ” . (1.4.2)
Taking p = b and q = 1 in Theorem 1.4.1 we obtain
C o ro lla ry 1.4.1. Let b be an integer > 1. Then
y ; ^ = - io g 6.r)
71=1
Corollary 1.4.1 is well-known. I t appears for example in [16, Problem 31],
[19], [22, p. 136].
We next use John’s theorem to evaluate the in fin ite series
^ (-1 )"-* flogn \ kn \ log 2 /
for any k E N. Recall th a t {y } = y — [y] is the fractional pa rt of the real
number y.
T heo rem 1.4.2. For k G N , we have
( ~ l ) n f logn V ___ 1_log 2.
Proof. In John’s theorem we take c = p/q = 2/1 and f { x ) = {a ;}fc (a; G
Then / is real valued on (—00, 00) and is periodic of period 1.
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CHAPTER 1. JOHN’S THEOREM 39
By (1.4.2) we have
a „(2) = —e„(2) = - ( - 1)» = ( - 1) " - 1.
Set
g(x) = xk, x e [0, 1],
, f s f 0, X e [0,1),h(x) = I
[ 1, s = 1.
Then g is o f bounded variation on [0,1] as i t is monotonic on [0,1] and h is
obviously of bounded variation. Now / = g — h so / is o f bounded variation
on [0,1] by Theorem 1.1.4.
Then, by John’s theorem, we have
= log 2 [ yk dy Jor yk+l 1 1
= log 2 1 y_k + l
1 log 2,k + 1
and the asserted result follows. □
Taking k ~ 1 in Theorem 1.4.2 we obtain
C o ro lla ry 1.4.2.
f^ (- l)" I logn 1 l , , . n§ n \ log 2 / 2 g '
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CHAPTER 1. JOHN’S THEOREM 40
Next we use John’s theorem to evaluate an in fin ite series which includes
° ° / 1 \7 1 r i _______ ^ kf s ( - l ) " f l o g n y n \ log 2 J
n = l
as the special case 6 = 2.
T h eo rem 1.4.3. Let k e N . Let b € N 6e such that b > 1. Then
OOy e J V r i o g r n i = _ _ l _ l o J
n 1 log 6 J Jfe + l &71=1 v '
Proof. In John’s theorem we take c = b. By (1.4.2), we see th a t
We choose f ( x ) = { x } k (x 6 R) so th a t / is real-valued on (—00, 00), is
periodic of period 1, and of bounded variation on [0, 1].
Then, by John’s theorem, we have
E T I 10* 1*71= 1 v
and the asserted result follows. □
°° € (b) f lo (ti/Qj] 1 ^F ina lly we use John’s theorem to evaluate the series > — ■ < —. , > .
£1 n \ log 6 /
T heorem 1.4.4. Let k € N. Lef a and b be integers with a > 1 and b > 1.
Then kr i o g ( n / q ) \ _ 1
^ n \ log 6 J fc + 1
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CHAPTER 1. JOHN’S THEOREM 41
Proof. We use John’s theorem in the form of Theorem 1.2.2. We choose
g(x) = { - x } k, c = b, * =
Then
®n(c) = €n(fi)
and John’s theorem gives
th a t is
^ n \ lo g t J k + 1 g ’
as asserted. □
1.5 The generalized Euler constants
Euler’s constant 7 is defined by
7 = lim [ V 1 - logn | = 0.5772156649.... (1.5.1)\ U 3 )
I t is well known th a t
^ i = log x + 7 + 0 ^ , (1.5.2)
as x —> +oo, see for example [3, p. 55].
The generalized Euler constants 7* (k = 0 ,1 ,2 , . . . ) are defined by
* = 0.1.2........ (1.5.3)n_>0° \ 3 k + 1 J
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CHAPTER 1. JOHN’S THEOREM 42
see [12, p. 259], where we understand log0 1 = 1. C learly
7o = 7* (1.5.4)
From [3, p. 70] we know th a t
£ ^ = i l 0 g’ * + 71 + 0 ( ^ ) , ( 1.5.5)
as x —> +oo.
n 2 V xn<x v
We next use the asymptotic formulas (1.5.2) and (1.5.5) to evaluate the00 jo jj
series ^ ( —l )n_1------- , which w ill be needed in the next section.n n = 1
Theorem 1.5.1.
E ( - l ) " - 1^ = —7 log 2 + log2 2., 71 71= 1
Proof. Let N € N. Then
2 N . 2 N , 2 N
= £ « - i7 1 = 1 7 1 = 1 71=1
IVo log 2n
^ 2n
log 2n
71=1
N
= -Ei n71=1
N
E log 2 + logn
i n71=1
logn- k « » E " E -n n
7 1= 1 71=1
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CHAPTER 1. JOHN’S THEOREM 43
Hence
Y ( - = V'!2«!!_log2y' I_y' ‘2«Iin -4-f n n n
n = 1 n= l 7i=l n=l
= 5 log2(2JV) + T1 + o ( ^ H )
— log 2 ( lo g N + 'y + O
- Q lo g > W + 11 + 0 ( ^ ) )
= Q (log 2 + log iV )2 - log 2 log N - i log2 N^j
+ (7 i “ 7 log 2 - 7 i) + 0
= Q log2 2 + log 2 log N + ^ log2 N - log 2 log iV - ^ log2 iV
-7 lo g 2 + o ( ^ )
= - log2 2 — 7 log 2 + O •
Letting N —> +oo we obtain
V " '(—l ) n -1 logn 1 2o---------------------- —7 log 2 + - log 2,
n 271=1
which gives the asserted result. □
Theorem 1.5.1 can be found in [12, p. 263] and [15, p. 288].
Our next result provides a generalization of Theorem 1.5.1.
T he o rem 1.5.2. Let b > 1 be an integer. Then
£ £n(5) = 7 log b - | log2 b.71= 1
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CHAPTER 1. JOHN’S THEOREM 44
Proof. For x > 1 we have
logn -<r-v m -,\l°g n v™' lognE « - w T = E f e w + D ^ - E - nn < x n<x n< x
- f t V ' lo g n _ y ' lo gnn n
n<x n<»xb]n
_ b y - log (fan) logn, bn " n
n<x/b n<x
= log6 Y ) i + E ! ^ - V l 2 L n“ n ' n “ n
n<«/fe n<x/b n<x
= lo g i, ( lo g f + 7 + o ( ^ ) )
- Q log2 x + 71+ )"logs
s= log b log s — log2 6 + 7 log b
~ log 6(2 log x - log b) + O
= “ log2 6 + 7 log 6 + O •
Letting x —> +oo we obtain
£ e” (6) ~ ~ = 7 log 6 ~ ^ log2 b.n= 1
□
Theorem 1.5.1 is the special case b = 2 of Theorem 1.5.2.
In 1972 Liang and Todd [24] proved the follow ing extension of Theorem
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CHAPTER1. JOHN’S THEOREM
n = l t>=0 ' '
We generalize Liang and Todd’s result as follows.
T heo rem 1.5.3. Let b > 1 be an integer. Let k e N. Then
^ /1Alog* n log^+ 'JE ‘ • w — = E L I (log6) “ t + T '7 1 = 1 V = 0 V '
Proof. Let N 6 N. Then
°° i fc , 1.en(6 )__ = hm 2^en(6)——
n at—»oo ' n71= 1 71= 1
{ JW 1 fc 1 «V ( 6 - 1 ) ! ^ + E ( - I ) ^ n
' n y nn = l n = l6|n tyVi
{ Mb 1 k Mb , ky » ^ - y y - D — ”
' n *—• nn = 1 n = lb|T.
{ IV 1 fc L 1 fc 'y - r log nb log n 2 - i n 2 ^ n7 1= 1 71= 1
— (logn + lo g 6) fc log* n
fc,log n
= nm yN-+00 n n
1 7 1= 1 71= 1
{ N 1 k / u \ Nb
7 1= 1 S=0 ' ' 71= 1
{ k / U \ N 1 « Nb 1 ft
S Q ^ E ^ - E ^ "5=0 x ' n=l n=l
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CHAPTER 1. JOHN’S THEOREM 46
log k n log k+1N b \ logfc+1 Nb
s—0
Nb
E n k + l / f c + lkTl=l
k fk= E L lo^ sb j s - i k
s= 0 ' S
N->oo \ E-/ \ s I S + l f c + l
■ 5 C R " ’-
k - i /jfcN
,• S 1 /, t , ,» M log^ 6 log^+ JV6'+ i t e S o ( r + r ( g g ^ “ T + i r + i - ,
- E(TW^»/lo g fc+1 Nb log*+1 b logk+l N b \
+ limN-* oo \ A; + 1 A; + 1 k + l
f k \ k_ log* * 16= E ( J l0 b ' i ' - ~ k + T -3 = 0
□
Theorem 1.5.3 can also be deduced from Corollary 1 and Proposition 4
in [12, pp. 260-261].
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CHAPTER 1. JOHN'S THEOREM 47
Before proceeding we recall Euler’s summation formula.
T h eo rem 1.5.4. I f f has a continuous derivative f on the interval [y,x],
where 0 < y < x, then
= / f ( t) dt + [ ( * - [ * ] ) / ' ( * ) <**y<n<x V y
+ / ( * ) ( N - x) - f(y)(ly) - v)‘
Proof. See [3, p. 54]. □
T heorem 1.5.5. Let k E N . Then
E log k n log k+1n f\o g k x '- Z — = 7. , , + 7fc + O
n<xn k + 1
as x oo.
Proof. In Theorem 1.5.4 we choose y = 1 and f ( x ) = ^ - £. C learly
k logfc_1 x — logfc x/ '(* ) = X*
is continuous on [ l,x ]. Then Euler’s summation form ula gives
l< n < x
k\ogk l t — logfet 'dt
Now
/
log XX
(N - x).
log Ktdt =
log k+1t k + l
x _ logfc+1 X
i k + 1
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CHAPTER 1. JOHN’S THEOREM 48
' k log t — log t
= / V w > (
J dt
k log*-1 1 - logk t st 2
dt
poo
/ (t-W)J %h\ogk l t — \ogKt '
a : t 2.fc- i o r t
t
— + O^log fcx '
for x sufficiently large and a constant Ak\ and
! ^ ([a;] _ x) = o ^X
Thus
En<x
log* n log*+1xn
+ Ak + O'\ogk x '
As
k + l
log* n logfe+1 x n k + 1
we deduce th a t Ak = 7&, so tha t
log* n log*+1x
lim f v 'x—*oo I / \ n < x
En< x
n k + l + 7k + 0
x
— Ik i
^ lo g *s '
dt
as asserted. □
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CHAPTER 1. JOHN’S THEOREM 49
1.6 Dr. Vacca’s series for 7
Euler’s constant 7 is defined by means o f a lim it, namely
7 = lim f 1 + 7H 1- - - lo g n ) .rc—>oo y i n J
Vacca [33] in 1909 was the firs t person to show th a t 7 can be expressed by
means o f an in fin ite series, namely
( - 1)'
71= 1n
lognlog 2
The series (1.6.1) has become known as “D r. Vacca’s series for 7 ” . We
obtain (1.6.1) by deducing i t from Corollary 1.4.2 and Theorem 1.5.1. For
other proofs o f (1.6.1), see Addison [1], Bauer [6], Gerst [13], Koecher [21],
and Sandham [31].
Theorem 1.6.1.( - 1)”
n = ln
logn log 2
Proof. As [y} = y — {y } , we have
£71=1
( ~ l) rn
logn log 2 E ( - 1) " /lo g n J 'lo g n 'lN
^ n \ l ° g 2 \ lo g 2 j ;
_ y > ( - l ) n logn y s ( ~ l) n f lo g n ) n log 2 n 1 log 2 J
n = l ° n = l v y^_-| 7»
lo§ 2 S n l0gn 5 n \ log 2 /_ ^ ( - 1)^ f lo g n )
71= 1
log 27 log 2 - ^ log2 2 ) - log 2
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CHAPTER1. JOHN’S THEOREM 50
by Theorem 1.5.1 and Corollary 1.4.2 respectively. Hence
E71= 1
( - i)"n
logn log 2
= 7 _ I l og2 + I lo g 2 = 7 ,
as asserted. □
As a consequence of Theorem 1.6.1 we have the follow ing alternative series
for 7 .
T heo rem 1.6.2.
2n+1 - 1
Proof. We have
y . / 1_ _ 1 _ 1 ,^ n \ 2 n 2n + 1 2n+1 - 1 n = l ' >
00 2n+1—1
E»En= 1 m=2n 00 2n+1- l
( - 1) 'm
= E En—1 m—2"
00 2n+1- l
( - 1) 'm
-n
n = 1 m =2n
00 -Qm
by Theorem 1.6.1. □
Our next result generalizes the Vacca series for 7 .
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CHAPTER 1. JOHN’S THEOREM 51
T heorem 1.6.3. Let b be an integer > 1. Then
00
7 = Een(6)
n = 1n
lognlog6
Proof. We have
E7 1= 1
Cn(6)n
logn log b
_ en(b) f log n f logn ) \ ^ n \ log 6 \ lo g fe /y
- OO 1 0 0
k E ^ - Elog 6
1log 6
7-
71= 1 71= 1
en(b) f log n n \ n ,
7 log 6 — i log2 6^ - ^ log b
□
Theorem 1.6.3 is due to Berndt and Bowman [8, p. 22, Theorem 2.6].
Our proof is much simpler than th a t of Berndt and Bowman, which involves
complicated integrals.
Exactly as we proved Theorem 1.6.2, we can prove the follow ing result as
a consequence of Theorem 1.6.3.
T heo rem 1.6.4. Let b be an integer > 1. Then
7 = ? _ + ________’ v bn bn + 1 bn+1 - I J 'n = l x '
The follow ing generalization of Theorem 1.6.3 was proved recently by
Berndt and Bowman [8, Theorem 2.8, p. 23].
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CHAPTER 1. JOHN’S THEOREM
T h eorem 1.6.5. Let a and b be integers with a > 1 and b > 1. Then
t = Een(&)
nlog(n /a)
log b
a—1 1
- lo§ a-71= 1
Proof. We have
En~a
£n(b)n
lo g in /a ) ' log b
en(b) / log (n /a ) f log(n /a ) } \^ n V lo g b I log b ] )
— _ 1 _ V f h ) lo g a £n(b) v e»(6) f log 6 " n log 6 " n n 1
° 7 1= 1 ° 7 1= 1 7 1= 1 V'
e»(6) f log(re/a) log 6
= ^ (^ log fe - | lo g 2 fc) - § | (- 'o s ^ - (- | I ° S &)
= 7 - ^ lo g 6 + loga + ^ log&
= 7 + log a.
Hence
7 - E= E
log (n /a) log b
— log a
log (n/a)log a.
71=1
Fina llya—1
E71—*1
(&)n
log (n /a ) log b
0—1
- E €n(b) + 1 log(n /a)
71= 1
a—1
n
£ -' nn —1b \n
lo g (n /a )’ log b ,
log 6a—1
a—1 ..
- E 1“ n71=1
lo g (n /a )' log b
U*—X -j
E 171=1
log (n /a )log 6
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CHAPTER 1. JOHN’S THEOREM 53
E 1\<n<a /b
log (bn/a) log 6
log bE H 1+\<n<a /b
E - - E -■4—', n ' n1 <n<a/b a/b<n<a
1 <n<a
log (n /a ) '
log (n /a) log b
E 1l< n < o
log(n /a) log 6
log (n /a )log b
E ; - E £(-1)~n<a/b
E -1 < n < a a —1 -
E v
, n ' n1 <n<a/b a/b<n<a
n= 1
□
Again our proof of Theorem 1.6.5 is simpler than th a t o f Berndt and
Bowman [8, Theorem 2.8]. Berndt and Bowman remark th a t th is theorem
is “apparently equivalent to ” a theorem o f Glaisher [14] w ithou t giving any
details.
In Theorem 1.5.3 we evaluated the in fin ite series
QO 1 kX y n( b ) - ^ , b (> 1 ) € Z , k e Kn = l
We now have enough inform ation to estimate the sum
6 ( > 1 ) € Z , k e N ,n<x n
fo r large x.
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CHAPTER 1. JOHN’S THEOREM 54
T heorem 1.6.6. Let b, k G N with b > 1. Then
5-0
as x —¥ oo.
Proof. Let b, k G N w ith b > 1, and rr G R. Then
m log *n V ^/1 , . V ' 1°gfenE £- ( 6) — = 2 ^ ( 1 + e* (6) )— - 2 . —n < s n<x n<x
— $ logfc n v logfc nn<* n < x6jn
_ logfc bn y-v logfc n“ n Z - j n
n<x/b n<x
_ (log6 + lo g n )fc y -v lo g fcnn n
n<x/b n<x
= E s e Q ^ + ^ - e ^71< x / 6 5 = 0 N 7
A:fcN\ i i - u r logSn v - ^ logfcn- „ "E „
«=0 ' / n<x/b n<x
( logfe+1 x/b ( logfc x1 7 + 7 k + 0 1 &
k + l \ xk
rn E (*+1)log‘~'6 log‘+1 x / h + ' E © logl5=0 x 7 5=0 x 7
logfc+1 x (\o g k xl k + 0
k + 1 \ x
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- sb 7*
CHAPTER 1. JOHN’S THEOREM 55
as asserted.
k +- ( lo g 6 + lo g x /b )k+l - — j— logfc+1 bi K + l
s = 0+E log*+1X
k + lJ k + O
'logk x 'X
- Es=0k
= E
logfe s 6 7S —
logfc-s ft 7s -s=0
k +
1k +
□
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Chapter 2
Radem acher’s Theorem
2.1 N otation
The follow ing notation is used throughout th is chapter.
Let K be an algebraic number field of degree n over the rational number
field Q. The ring of integers of K is denoted by O k - The number o f roots
o f un ity in K is denoted by w {K ), the discrim inant of K by d (K ) and the
regulator o f K by R {K ). The number o f real fields among the conjugate
fields o f K is denoted by r and the number o f nonreal fields by 2s so th a t
n = r + 2s. The structure constant of the fie ld K is the quantity
2 '+ V R ( K )
w (K )^ \d (K Y \'
Two nonzero ideals A and B o f Ok are said to be equivalent, w ritte n A ~ B,
56
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CHAPTER 2. RADEMACHER’S THEOREM 57
i f there exist 0) £ Ok and /3 (^ 0) £ Ok such tha t
< a > A —< (3> B. (2.1.2)
Clearly ~ is an equivalence relation on the set of nonzero ideals of Ok - The
equivalence class containing A is denoted by [A]. Since the quotient field of
Ok is K [2, Theorem 6.1.5, p. I l l ] , (2.1.2) is equivalent to the existence of
c(y^ 0) £ K such th a t
A = cB. (2.1.3)
The norm of an element c £ K is denoted by N(c) [2, p. 222] and the norm
of an ideal A o f Ok by N (A ) [2, p. 143]. The greatest common divisor of
A and B is denoted by (A, B ). The follow ing result w ill be im portant in the
extension of John’s theorem to algebraic number fields due to Rademacher
[27, Theorem, p. 173].
T heo rem 2.1.1. Let c 0) £ K . Then there exist unique nonzero ideals
A and B o f Ok such that
A = cB, (A, B ) = < 1 > .
Proof. As c 0) £ K there exist a { ^ 0) £ Ok and b £ N such th a t
c = a/b,
see for example [2, Theorem 4.2.6, p. 85]. Let P i, . . . ,P m be the set of
prime ideals which divide either < a > or < b > (or both). Then there exist
nonnegative integers a i , . . . , om and nonnegative integers b\ , . . . , bm such tha t
< a > = P 1a i- - -P f lm, < b > = P * 1 ■ • • P ^m.
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CHAPTER 2. RADEMACHER’S THEOREM 58
Reorder P i, . . . , Pm so th a t a* > bi fo r i = 1 ,2 , . . . , k and a; < bi for i =
k + 1, . . . ,ra . Set
= p ai-6i . . . p ak—bk Q _ p fck+l- “fc+i . . . p bm—am1 fe Aj+1 wi
Then
■Pi01 • • • pmm = < & > = < be > = c < b > = cPx61 • • • P ^m
soD ai-6l p Ofc—6fc - p 6fc+l-Ofc+l P b r r x - O m
1 ' " k k+l m >
tha t is,
A = cB.
Since the only prim e ideals d ivid ing A are P i , , Pk, the only prime ideals
d ivid ing B are Pk+i, . . . ,P m, and P i, . . . ,Pm are d istinct, i t follows tha t
(A ,B ) = < 1 > ,
which establishes the existence o f A and B.
Suppose A! and B ' are nonzero ideals o f Ok such tha t
A! = cB', (A', B ') = < 1 > .
Then
A 'B = (cB ')B = (<zB)B ' = A B '.
Hence
A I A 'B .
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CHAPTER 2. RADEMACHER’S THEOREM 59
B ut {A, B ) = < 1 > , so
A \ A ' ,
say
A ' = A M
for some ideal M o f Ok - Then
A M B = A 'B = A B ',
so
B ' = B M .
Hence
< 1 > = (A1, B ') = {A M , B M ) = M {A , B ) = M < 1 > = M .
Thus
A! = A < 1 > = A, B ' - B < 1 > = B ,
which establishes the uniqueness of A and B. □
E xam p le 2.1.1. Let K = Q (> /=5). We choose c = € K . We
determine ideals A and B o f Ok = Z + Z \ /—5 such that
A = ( 1 + ) B, { A ,B ) = < 1 > .
We /iave
N (1 + a /T 5) = 6 = 2-3, iV(2) = 4 = 22.
The prime ideal factorizations o f < 2 > and < 3 > are
< 2 > = P 2, < 3 > = PXP2,
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CHAPTER 2. RADEMACHER’S THEOREM 60
where
P = < 2,1 + V --5 > = < 2,1 - > ,
P i = < 3,1 + >/=5 > , P2 = < 3 , 1 - V C 5 > ,
see /o r example [2, p. 265]. Clearly P ^ P1; P ^ P2, P i ^ P2, N (P ) = 2
and JV(Pi) = iV(P2) = 3. Now
< l + v/= 5 > = < 1 + vc 3 > < 1 - V c 5,2,3,1 + >/Z5>
= < 6,2(1 + \ / —5), 3(1 + V —5), (1 + V - 5 ) 2 >
= < 2,1 + >/—5 > < 3,1 + V -5 >
- P P i
and
< 2 > = P 2,
so we choose (guided by the choice in the proof o f Theorem 2.1.1)
A = P l, B = P.
Clearly (A, B ) = 1 and
cB = 1 = < 1 + \ / —5 > < 2 > " * BZ
= P P \P ~2P = P i = A.
I f c e P - is such th a t jiV(c)| > 1 so th a t c ^ 0, by Theorem 2.1.1 there
exist unique nonzero ideals A and B o f Ok such tha t
A = cB, (A, B ) = 1,
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CHAPTER 2. RADEMACHER’S THEOREM 61
and we define by analogy w ith (1.2.2) the arithm etic function a/(c) for any
nonzero ideal I o f Ok by
af (c) = <
0, i f A / I , B / I ,
-N (A ) , if A 11, B / I ,
N (B ), i f A / / , B \ I ,
N ( B ) - N ( A ) , i f A \ I , B 11.
(2.1.4)
2.2 Rademacher’s extension of John’s
theorem
The follow ing extension of John’s theorem was proved by Rademacher [27,
Theorem, p. 173] in 1936.
T heorem 2.2.1. Let M be a nonzero ideal o f Ok - Let c € K be such that
|iV(c)| > 1. I f f ( x ) is Riemann integrable on [0,1] and o f period 1, then
where the sum is over nonzero ideals I o f Ok equivalent to M and the sum
mands are arranged according to increasing N ( I ) .
The above series inherits its convergence from the ordering o f N ( I) . Note
also th a t we only require the function f { x ) to be Riemann integrable on its
period, which is a looser requirement than being of bounded variation.
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CHAPTER 2. RADEMACHER’S THEOREM 62
2.3 Rademacher’s theorem for algebraic
number fields
In th is section, we consider the special case of Theorem 2.2.1 when f ( x ) is
identically 1 and K is assumed to contain an integral element c of norm ±2.
The la tte r condition ensures th a t a j(c) = ±1 for every nonzero ideal I o f
T heo rem 2.3.1. Let K be an algebraic number field such that there exists
c G Ok with |iV(c)| = 2. Let M be a nonzero ideal o f Ok - Then
where the sum is over a ll nonzero ideals I o f Ok equivalent to M and the
summands are arranged according to increasing N ( I ) , and
O k -
(2.3.1)-1 , i f < c > \ I .
Proof. We take
A = < c > , B = < 1 >
so that
c = A /B , (A, B ) = 1.
Also
N (A ) = N (< c > ) = \N(c)\ = 2.
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CHAPTER 2. RADEMACHER’S THEOREM 63
For any nonzero ideal I o f Ok we have
N ( B ) = 1,o j(c) = <
if < c > J ( I ,
N (B ) - N (A ) = 1 - 2 = - 1 , i f < c > | I ,
= —e(c, A).
Also let f ( x ) = 1 (x 6 R). Then f ( x ) is Riemann integrable on [0,1] w ith
Jq f ( x ) dx = 1 and has period 1 as f ( x + 1) = 1 = f ( x ) ( i e R). Hence, by
Rademacher’s theorem, we have
ai{c)E N ( I)= k log 2
so that
E
le [M ]
e(c, I ) 2T+snsR (K )
m m N W w ( K ) J \ m j \
completing the proof.
log 2
□
2.4 Rademacher’s theorem for imaginary
quadratic fields
Let K be an im aginary quadratic field. Then there exists a unique squarefree
integer m < 0 such th a t K = Q (y/m ), see for example [2, Theorem 5.4.1, p.
95]. In th is case
3 II r = 0, 3 = 1 ,*
2, i f m 7 —1, —3,
II 4, i f t o = — 1 ,
6, i f to = —3,
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CHAPTER 2. RADEMACHER’S THEOREM 64
d(K) =m, if to = 1 (mod 4),
0K =
4to, if to ^ 1 (mod 4),
Z + Z a/ to, ifT O = l (mod 4),
Z + , i f to ^ 1 (mod 4),
R ( K ) = 1,
see for example [2, Theorem 5.4.3, p. 98; Theorem 5.4.2, p. 96; D efin ition
13.7.1, p. 380]. The structure constant « of K is given by
7r 4 ’
7T
3 7 3 ’7r
v W ’7T
^ • y /H
if TO = —1,
i f to = -3 ,
i f to = 1 (mod 4), to ^ —3,
, i f to ^ 1 ( m o d 4 ) , T O ^ — 1.
(2.4.1)
Rademacher’s theorem for im aginary quadratic fields, a special case of
Theorem 2.3.1, is as follows:
T heo rem 2.4.1. Let K be an imaginary quadratic field such that there exists
c E Ok with N (c) = 2. Let to be the unique squarefree negative integer such
that K = Q (v/ro). Let M be a nonzero ideal o f Ok - Then
e(c, I )Eie[M] m
= —/clog 2,
where the sum is over nonzero ideals I o f Ok equivalent to M and the sum
mands are arranged according to increasing N ( I ) , e(c, I ) is given by (2.3.1)
and k is given by (2.4.1).
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CHAPTER 2. RADEMACHER’S THEOREM 65
We next determine those im aginary quadratic fields K such th a t O k
contains an element c of norm 2.
I f K = Q (\/m ) w ith m ^ 1 (mod 4) we seek c = x + y \/rn (x, y G Z)
such th a t a;2 + |to|t/2 = 2. I f |m| > 3 there are no integers x and y satisfying
th is equation. I f |m| = 2, th a t is m = —2, we can take x = 0, y — 1. I f
\m\ = 1, th a t is m = —1, we can take x =■ y — Thus we have the two
possibilities
K = Q { y / - i ) , c = 1 + *,
K = Q(y/^2), c= V = 2 .
I f K = Q (y/m ) w ith m = 1 (mod 4) we seek c = ( j j e Z , x =
Cu -4” Imhy2?/ (mod 2)) such th a t -------^—!— = 2, th a t is a:2 + \m\y2 = 8. I f \m\ > 9
there are no integers x and y satisfying th is equation. For \m\ < 8 the
eligible m = 1 (mod 4) are m = — 3 and m = —7. I f m = — 3 the equation
x2 + 3y2 = 8 has no solutions in integers x and y. I f \m\ = —7 the equation
x2 + 7y2 = 8 has the solution x — y = 1. This gives the single possibility
K = Q (v^7 ), c = i i | E Z .
We examine these three possibilities in the next three subsections.
2.4.1 K = Q (v/Z l) , c = 1 + i
W ith K = Q (a /- I) , we have Ok = Z + Z \ / —1 = Z + Z i As h (K ) = 1,
see for example [10, p. 151], Ok is a principal ideal domain (in fact, Ok is a
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CHAPTER 2. RADEMACHER’S THEOREM 66
Euclidean domain, see for example [2, p. 33]). Here by (2.4.1) the structure7T
constant is k = —.4
We choose
c = 1 + i
so that
N(c) = (1 + *)(1 - i ) = 2.
Thus for an a rb itra ry ideal I o f Ok = Z + Z i we have
, 1, i f < l + i> | I ,e (c ,I) =
-1, i f < 1 + i > / / .
We next determine a necessary and sufficient condition for an ideal I o f
Ok to be divisib le by < 1 + i > . As Ok is a principal ideal domain, we have
I = < a + b i> for some a,b 6 Z. Then
< 1 + i > 11 <=>■ < 1 + i > |< a + b i>
l + i \ a + bi
3 c, d € Z such tha t a + bi = (1 + i) (c + di)
•<==> 3 c, d 6 Z such th a t a = c — d,b — c + d
<==> a = b (mod 2).
Thus
e(c, / ) = e (l 4- i, < a + bi > ) = (—l ) a+6.
As h (K ) = 1 every nonzero ideal of Ok is equivalent to < 1 > so we choose
M = < 1 >
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CHAPTER 2. RADEMACHER’S THEOREM 67
so th a t w ith K * = i^ \ { 0 }
[M \ = {cxM C 0 K \<* e K * }
= {(a + bi) < 1 > C 0 K \a, b e Q, (a, b) ^ (0 ,0 )}
= { < a + bi > \a,b e Z , (a, b) ^ (0 ,0 )}.
Hence, by Theorem 2.4.1 we have
V ' (Z 1.)!+b = —— log 2a2 + b2 4 g
/= < a + 6 i>a,6eZ
Now
< a + > = < a' + b'i > a + bi = 6 (a' + b 'i),
for some u n it 0 o f Ok- Since the only units in O k are ±1 , ± i, we deduce
th a t(—i)o+b ^ x (-!)«+ »
a2 + 62 4 a2 + 627^0 a,bGZ
I= < a + b i> (a,fc)^( 0,0)QttbGZ
Hence we have proved the follow ing result.
Theorem 2.4.2.( _ ! ) « *
4 - ; ^ - m 2 8(a, 6)^(0,0)
We emphasize th a t the sum in Theorem 2.4.2 is ordered according to
increasing values of a2 + b2 so tha t
4 4 4 8 4 4 8 ,“ I + 2 + 4 “ 5 + 8 “ 9 + l 0 ------------~ W,0g2-
As a check on our calculations, we derive Theorem 2.4.2 in another way.
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CHAPTER 2. RADEMACHER’S THEOREM 68
Alternate proof o f Theorem 2.4.2. We have
(_ i)°+ » “ „ ( - i ) a+62 ~* a2 + b2 2—i 2-4 a2 + b2
a,beZ n = l a,6GZ(a,b)^(0,0) a2+b2= n
= E in—1 a,6GZ
a2+62= n
as 2 i l2a + b = a + b — n (mod 2).
Now, by a classical result (see for example [17, pp. 115-120], [35]), we have
E i=*£/-4'(
a2
so tha t
X da,bSZ d\na?+b2=n
2 -~> a2 + b2 2— 1 n 2—i I da,beZ n = l d |n
(<z,6) (0f0)
mmd =l e = lVd
, < l 1 \ A 1 1 1- 4 1 - - + - ------- 1 - X + X - T +3 5 J \ 2 3 4
= “ 4 ( 0 (log 2)
= -7T log 2.
as desired. ^
We note th a t Theorem 2.4.2 agrees w ith the fina l form ula in [27] and w ith
the value of 02(1) given in [37, p. 192].
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CHAPTER 2. RADEMACHER’S THEOREM 69
2.4.2 K — Q ( \ / -2 ) , c = V ^ 2
W ith K = Q ( \/—2), we have Ok = Z + Z \ /—2- As /i(A ') = 1 (see for
example [10, p. 151]), O# is a principal ideal domain (in fact, Ok is a
Euclidean domain, see for example [2, p. 33]). Here by (2.4.1) the structure7T
constant is re = — ■=.2 y/ 2
We choose
c = V —2
so tha t
7V(c) = (> /= 2 )(-V = 2 ) = 2.
Thus for an a rb itra ry ideal J o f Ok = Z + Z \ /—2, we have
1 ,if < \/~ 2 > |
1, i f < y p 2 > / I .
We next determine a necessary and sufficient condition for an ideal I o f
O k be to divisible by < y/ — 2 > . As O k is a principal ideal domain we have
I = < a + b^f—2 > for some a, b e Z. Then
< y f— 2 > | / 4=4> < \ /^ 2 > |< a + b \ fm >
y/ — 2 | a + by/ —2
>/—2 | a
4=r> 2 | a.
Thus
e(c, / ) — e (\/—2, < a + by/ ^ 2 > ) = ( -1 ) “ .
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CHAPTER 2. RADEMACHER’S THEOREM 70
As h (K ) = 1, every nonzero ideal of Ok is equivalent to < 1 > so we
choose
M —< 1 >
so tha t
[M \ = {a M C Ok \ol G K * }
= {(a + by/^2) < 1 >C 0 ^ |a ,6 G Q, (a,b) / (0 ,0 )}
= {< a + &>/—2 > |a, 6 G Z, (a, 6) ^ (0 ,0 )}.
Hence, by Theorem 2.4.1, we have
/#< o>/=<a+6's/~2>
a,6€Z
(~1)° = l _ l 0g2^ a2 + 2 b2 2 ^2
Now
< a + by/— 2 > = < a7 + b'yf— 2 > <£=>■ a + by/ — 2 = 6 {a! + 6z\Z~2),
for some u n it 0 o f Ok = Z + Z \ /—2- But the only units o f Z + Z%/—2 are
±1 so
V ( - 1)* 1 V ( - 1 ) °
. A ; a2 + 2 ^ 2 4 - i a2 |2 6 2'/#< 0> a,6ez/=<a+fcV—2> (a,6) (0,0)
a,b€Z
Hence we have proved the follow ing result.
T heo rem 2.4.3.
(a ,6 ) /(0 ,0 )
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CHAPTER 2. RADEMACHER’S THEOREM 71
As a check on our calculations, we derive Theorem 2.4.3 in another way.
Alternate proof o f Theorem 2.4.3. We have (as a = a2 = a2+2&2 (mod 2))
(-1)° = ~ v (~l)a4"1 a2 + 262 " 4 - a2 + 262
a,i>6 Z n = l a,6GZ(a,6)^(0,0) a2+262= n
= E ^ r E i-n = l o,6GZ
a2+262=7i
Now
E ! = 2E (t )a,6eZ d|n ' '
a2+2b2=n
(see for example [11, Theorem 64, p. 78], [36]) so th a t
( - l ) r t * = ~ ( - i ) . / _ 8 -
+ 262 Z -. n d .(
(_ l)de / _8
a,6G Z 7 i= l d|n(a, 6)^(0,0)
^ ^ de V da,e= l N
= 2 E( - l ) de / —8'
de \ d Ja ,e= l x '
-e = l d = l
= -2 (lo g 2 ) L ( l, -8 ) ,
where the D irich le t L-series L ( l, D ) for an a rb itra ry discrim inant D is given
by^ ( - ) t(l,D) = E iJ2-n ~ \
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CHAPTER 2. RADEMACHER’S THEOREM 72
For D < 0 by D irich le t’s class number form ula we have
■ A (g ) 2tth(D )^ n w (D )-J \D \'
see for example [17, Theorem 10.1, p. 321]. W ith D = -8 , as w (—8) = 2,
\J \D \ = y/ 8 = 2y/2 and h (—8) = 1, we have
f ' S ) = - J Ln 2^2"71=1 V
Thus
T (~ 1}‘ - 2floC2) * - ’rl0g2 ** + » 2v*2 V2 '
(a,6 )^ (0 ,0)
We note th a t Theorem 2.4.3 agrees w ith the value of 0 i( \/2 ) given in [37,
p. 192].
W ith K = Q C v^T), we have 0 K = Z + Z ( i± ^ ) . As h (K ) = 1 (see
for example [10, p. 151]), Ok is a principal ideal domain (in fact, Ok is a
Euclidean domain, see for example [2, p. 34]). Here by (2.4.1) the structure7T
constant is k = —=.V7
We choose1 + a/= 7
C = ^ —
so tha t
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CHAPTER 2. RADEMACHER’S THEOREM 73
Thus for an a rb itra ry ideal I o f Ok = Z + Z ^ 14Y ~ ^ , we have
l , i f ( i ± # z ) | / ,e(c, I ) =
_ l , i f
We next determine a necessary and sufficient condition for an ideal I o f
Ok to be divisib le by *s a Pr inciPal ideal domain we
have I = (^a + b |
<i±r EZ>
1 +' j for some a, b G Z. Then
( !± * 2 >1 + y/—7
21 + y/ ~ 7
a + b
• * * m >
t 3 )
1^3) (1^3)
2 I a.
a
a
Thus
e (c ,/) = e
As h (K ) = 1, every nonzero ideal of Ok is equivalent to < 1 > so we
choose
M = < 1 >
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CHAPTER 2. RADEMACHER’S THEOREM 74
so tha t
As
[M \ = {a M C Ok W € K * }
= {(z + y V ^ f ) < 1 >C 0*1®, y e Q, (x, y) ^ (0 ,0 )}
= | / a + 6 f l ^ - I ) \ k 6 e Z , ( a , 6 ) ^ ( 0 , 0 ) j .
= ( a + & ( i J ^ E Z ) ) (a+&(lz EZ))— o? 4* ob -4- 2 ,
by Theorem 2.4.1 we have
/ - iv *= — 7= log 2.y - ( ~ l ) a ____
a2 + ab + 2 b2 y/ 7i^< o>
a,&€Z
Now
„ . ( l ± £ 1 ) .
where 0 is a u n it o f Ok = Z + Z . B ut the only units o f Z + Z
are ±1 so
v ( ~ l) a _ 1 v (~ 1)arT~Z a2 + ab + 2 b2 2 ^ a2 + ab + 2 b2'/#< o> a,bez
J = ( a + b ( ^ V Z?) ) (o.6)#(0,0)a,e>€Z
Hence we have proved the follow ing result.
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CHAPTER 2. RADEMACHER’S THEOREM 75
T heorem 2.4.4.
V ' (~ 1}° - log 22 , a’ + a& + 26> V7
(a,6) (0,0)
As the pa rity o f a is not d irectly related to the parity of a2 + a6 + 262, i t is
not possible to give a proof o f Theorem 2.4.4 along the lines of the alternative
proofs of Theorems 2.4.2 and 2.4.3 given earlier.
2.5 Rademacher’s theorem for real quadratic
fields
Let K be an real quadratic field. Then there exists a unique squarefree
integer m > 0 such th a t K — see for example [2, Theorem 5.4.1, p.
95]. In th is case
n = 2, r = 2, s = 0,
w (K ) = 2,
{ m, i f m = 1 (mod 4), v
4m, i f m ^ 1 (mod 4),
{Z + Z yfrri, i f m = 1 (mod 4),
Z + z ( ± ± ^ ) , i f m ^ l (mod 4),
R (K ) = log rj, where 77 is the fundamental u n it(> 1) o f AT,
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CHAPTER 2. RADEMACHER’S THEOREM 76
see for example [2, Theorem 5.4.2, p. 96; Theorem 13.7.1, p. 380]. Thus, as
Rademacher’s theorem fo r real quadratic fields, a special case of Theorem
2.3.1 is as follows:
T heo rem 2.5.1. Let K be a real quadratic field such that there exists c & Ok
with |iV(c)| = 2. Let m be the unique squarefree positive integer such that
K = Q (y/m ). Let M be a nonzero ideal o f Ok - Then
where the sum is over nonzero ideals I o f Ok equivalent to M and the sum
mands are arranged according to increasing N ( I ) , e(c, I ) is given by (2.3.1)
and n is given by (2.5.1).
D e fin itio n 2.5.1. An integer a o f areal quadratic field is said to be prim ary
the field.
T h eorem 2.5.2. Every nonzero real quadratic integer has exactly one asso
ciate which is primary.
i f m ^ 1 (mod 4).
(mod 4)(2.5.1)
a > 0, I < —7 < r f , a
where a ' denotes the conjugate o f a and rj (> 1) is the fundamental un it o f
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CHAPTER 2. RADEMACHER’S THEOREM 77
Theorem 2.5.2 is proved in Cohn [10, Theorem 3, p. 146].
In Section 2.4 we saw th a t there are only three im aginary quadratic fields
K containing an integral element c o f norm ±2. In contrast each of the
in fin ite ly many real quadratic fields Q(y/p) (p (prime) = 3 (mod 4)) contains
such an element.
2 .5 .1 K = Q ( \ /2 ) , c = a/2
W ith K = Q (\/2 ), we have Ok = Z + Z \/2 - As h (K ) = 1 (see for example [10,
p. 271]), Ok is a principal ideal domain (in fact, Ok is a Euclidean domain [2,
Theorem 2.2.6, p. 35]). The fundamental u n it is ift = l+ \ /2 (see for example
[2, Example 13.7.1, p. 381]), so the regulator is R (K ) = lo g (l + \/2 ). Here
by (2.5.1) the structure constant is k =
We choose
c V2
so tha t
N(c) = (V2)(-\/2) = -2.
Thus for an a rb itra ry ideal I o f Ok = Z + Z \/2 , we have
eM)J i . * < V 2 > | / ,
\ - l , i f (V2)XI .
We next determine a necessary and sufficient condition for an ideal I o f
Ok to be divisib le by (\/2 )- As Ok is a principal ideal domain we have
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CHAPTER 2. RADEMACHER’S THEOREM 78
I = < a + by/2 > for some a, 6 6 Z. Then
< V 2 > \ I <=$> < V 2 > \< a + by/ 2 >
y/ 2 | a 4- by/ 2
V 2 | a
2 I a.
Thus
e(c, / ) = e(V2, < a + 6\/2 > ) = (—l ) a.
As h (K ) = 1, every nonzero ideal of Ok is equivalent to < 1 > so we
choose
M = < 1 >
so tha t
[M ] = {a M C O k\& € AT*}
= {(a + &V2) < 1 > C 0 K\a, b € Q, (a, 6) / (0 ,0 )}
= | < a + by/2 > |a, 6 € Z, (a, b) / (0,0). jAs N (< a > ) = |Af(aOI> we have
N ( < a + by/2 > ) = |JV(a + &V2)|
= | (a + 6 \/2 ) (a — by/2) |
= |a2 — 262|,
and so by Theorem 2.5.1, we have
V ,(-!)“ = _ M 1.±.,V2)1 2l“ 2 - 2^1 V2 l0g2-
/=<a+b\/2>a,beZ
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CHAPTER 2. RADEMACHER’S THEOREM 79
We next determine a unique generator a+by/2 for the ideal I . By Theorem
Let p (prime) = 3 (mod 4) and le t K = Then Ok = Z + Z^/p.
We note th a t Ok is not necessarily a principal ideal domain. Here we have
n = 2, r = 2, s = 0, w (K ) = 2, d (K ) = 4p and R (K ) = log rj, where rj is the
fundamental u n it of K .
I f p — 3 (mod 8) then x2 — py2 = - 2 is solvable in integers x, y, and if
p = 7 (mod 8) then x2 —py2 = +2 is solvable in integers x, y, see for example
[2, Exercises 14,15, p. 297] respectively. This gives rise to an element o f norm
(—l ) E7i 2 in K = Q (y/p) whenever p = 3 (mod 4). Let c = x + y^Jp £ O k
2.5.2 we set a + by/2 in the range
a + W 2 > 0 , 1 < a + b ^ < ( 1 + a / 2 ) 2 ,a — by/ 2
th a t is,
a + bV2 > 0, 1 < < 3 + 2 ^ ,a — by/ 2
to get a unique generator.
Hence we have proved the follow ing result.
T heo rem 2.5.3.
a,be Za+b'v/2>0
2.5.2 K = Q (y/p), p (prime) = 3 (mod 4)
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CHAPTER 2. RADEMACHER’S THEOREM 80
be such an element of norm (—1 )^ 2 . As x2 — py2 = (—1)£412 we have
x2 + y2 = 2 (mod 4) so th a t x = y = 1 (mod 2).
Let I be a nonzero principal ideal of Ok - Then, by Theorem 2.5.2, there
exists a unique element a + by/p o f Ok such th a t I = < a + by/p > and
a + by/pa + by/p > 0 , 1 <
We have
e(c, I ) = e(c, < a + by/p > ) = <
Now, as x = y = 1 (mod 2), we obtain
a - b y / p < V
1, i f c | a + by/p,
- 1 , i f c / a + by/p.
so tha t
Also
c | a + by/p 4=» x + yyfp | a + by/p
2 | (a + by/p)(x - yy'p)
2 | (az - pby) - (ay -
4= ^ 2 \ (a + b) - (a + b)y/p
2 I a + b
e(c, / ) = e(x + y y /p , < a + by/p > ) = ( - l ) a+6.
N ( I ) = N ( < a + by/p > ) = |iV(o + by/p) \
= \ (a + b y / p ) ( a - b y / p ) \ = \a2 - p b 2 \.
Hence, by Theorem 2.3.1 we have
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CHAPTER 2. RADEMACHER’S THEOREM 81
T heorem 2.5.4.
(_l)a+f> (log 2) (log rj)
S I VPa+by/p> 0
2.6 Rademacher’s theorem for real cubic fields
with two nonreal embeddings
Let K be a real cubic field w ith two nonreal conjugate fields. Then n = 3,
r = 1 and s = 1. The only roots of un ity in K are ±1 so w (K ) — 2 (see
for example [2, Theorem 13.5.3, p. 367]). The cubic field K has a unique
fundamental u n it 77 (> 1) (see for example [2, Theorem 13.4.2, p. 362]). The
regulator R {K ) of K is
R (K ) = | log 7 7] = log 77,
(see for example [2, D efin ition 13.7.1, p. 380]). The structure constant o f K
2r+BirsR (K ) 2tt log r/
w (k ) V \ W T \ V W M '
T heorem 2.6.1. Let K be a real cubic field with two nonreal embeddings.
Letr) > 1 be the unique fundamental un it o f Ok , so that a ll units o f Ok are
given by ± 77™ (71 € Z ). Given a £ Ok \ { 0} there exists a unique associate f i
o f a such that
13 > ° ’ 1 ~ \NQ3)\V2 < V'
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CHAPTER 2. RADEMACHER’S THEOREM 82
Proof. The associates o f a are ± rjna (n € Z ). I f a > 0, we choose /3 =
+r]na > 0, and i f a < 0, we choose (3 = —r)na > 0. Thus /3 is a positive
associate of a.
Clearly
P = \P \ = Vn\<*\ = sgn(a)77n.
Denoting the conjugates o f (3 by 0 and 0 ' , we obtain
0 = ± (V ) a , 0 ' = ± a .
Hencea' a" ( ' " \ n ' "f3 (3 = ( e e ) a a .
As 77 is a u n it o f Ok we have
777/ 77” = A (77) = ±1.
Now 77 > 1 and 77" = r f so th a t 0 0 ' = 0 0 = \0 12 > 0 and thus
' " 17777 77 = 1.
Hence
Thus
and so
/ / / 1 ” = v
77n
pprpr
P \a\rj2n
\P'P"\ K aw|
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CHAPTER 2. RADEMACHER’S THEOREM 83
|o;|7y2rlNow ; „, is a s tric tly increasing function of n G Z as u > 1 such tha t
|o; a I
.. \a\rpn \a\rj2nhm \ - r - a 7 = 0. 1™ Tr~ n7 = + 00.n~ *~ 00 |q: a I n-»+oo | a a |
Hence there exists a unique m G Z such tha t|o ; |^ 2 ( m - l ) ^ ^ |o ; |^ 2m
th a t is
/ 111 ^ l — i / n 1 >a a cc a
| a aThus 0 = sgn(a)r}n is the unique associate of a such tha t
00>O, 1 <F ina lly we note tha t
0 0
01 0
< 1 f .
so th a t the condition 1 <
0 0 " 0 0 0 " N (0 )
- K < rj2 is equivalent to
1 <~ \N (0 )\W V
giving the asserted result. □
T h e o rem 2.6.2. Let K be a real cubic field with two nonreal conjugate fields.
Suppose that there exists c G O k with |iV(c)| — 2. Then
e(c, I ) 27r(log7?)(log2)
V N ( ! ) s / W ) '
where the sum is over a ll nonzero principal ideals I o f Ok and the summands
are arranged according to increasing N ( I ) , and e(c, I ) is given by (2.3.1).
We now apply Theorem 2.6.2 to the cubic fie ld K = Q (v/,2) which is
a real cubic fie ld w ith two nonreal conjugate fields Q (w s/2) and Q(tu2s /2),
where w G C is such th a t w ^ 1, w3 = 1.
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CHAPTER 2. RADEMACHER’S THEOREM 84
2.6.1 K = Q ( \//2), c = \/2
Let K = Q (s/2). Then d (K ) = —108, see for example [2, Table 1, p. 177].
As h (K ) = 1 (see for example [2, Table 9, p. 329]), O k = {u + v \ / 2 +
vj{y/2)2 \u ,v ,w € Z } is a principal ideal domain. Let c = \/2 G Ok- Then
N(c) = 2. We have n = 2, r = 1, a = 1, w (K ) = 2, d {K ) = -108. The
fundamental u n it is rj = l + \ / 2 + ( \ /2 ) 2 [2, Table 11, p. 375], so the regulator
is R {K ) = lo g (l + $ 2 + (V 2 )2). Hence
27rIog(l + ^ + ( ^ ) 2) 7rlog(l + V 2 + ( V 2 ) 2)K V I -1 0 8 ] 3 \/3
Let I be a principal ideal of Ok - Then, by Theorem 2.6.1, there exists a
unique (5 = x + y \ / 2 + z(ty2)2 G Ok such th a t I = < (3 > and
/ 3 > 0 , 1 - |iV(/?)|V2 < r >
th a t is,
x + y + z{\J2)2 > 0 , 1 <x + y V2 + z { $ 2 ) 2
< 1 + V2 + (v ^ )(x3 + 2 y3 + 4 z3 — 6 xyz ) 1! 2
as
N ^ + y y f i + z \ f 2 = a:3 + 2y3 + 4z3 — 6 xyz > 0.
We have
e(c,J) = e (c ,< /3 > )
1, i f < c > |< j3 >
-1 , i f < c > /< 0 >
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CHAPTER 2. RADEMACHER’S THEOREM 85
= ( - l ) x-
Hence, by Theorem 2.3.1 we have
T heo rem 2.6.3.
( -1 ) *
1, i f c | 0 ,
-1 , i f c //3 ,
1, i f v^2 | x + y f f i + z ( ^ 2 )2,
-1 , i f y/2 )(x + y ^ / 2 + z ( ^ 2 )2,
1, i f v^2 | x,)
-1 , i f \^2 )(x,1
1, i f 2 | x,
-1 , i f 2 /x ,
E „ x 3 + 2 y3 + 4 z3 — 6 xyzx,y,z€Z y y
x+yfy2+z{fy2)2>0
7r(log 2) lo g (l + y/2 + ( y ^ )2)
3 \/3
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Chapter 3
Kronecker’s Theorem
3.1 Dedekind eta function
In th is section we give some properties of the Dedekind eta function th a t we
w ill need in the next section.
Let H denote the upper ha lf of the complex plane, th a t is
H — {z G C\z = x + iy , x ,y eM , y > 0}.
For z e H the Dedekind eta function 77(z) is defined by
OO
ij(z ) = eKiz/12 [ I ( ! “ e2” imz). (3.1.1)771=1
This in fin ite product converges absolutely and uniform ly in every compact
subset of H , see for example [32, p. 15]. Thus i](z) is analytic in H . Since
86
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CHAPTER 3. KRONECKER’S THEOREM 87
none of the factors o f the convergent in fin ite product is zero for z € H it
follows th a t r)(z) ^ 0 for z € H , see [32].
Theorem 3.1.1.
\rj{x + iy)\ = \r } ( -x + iy)\
fo r all x, y € M with y > 0.
Proof. Let x, y G K. w ith y > 0. Then
OO
r)(±x + iy ) = J J ( l - e2nim ±x+i^ )m= 1
oo= n a - e~2nmye±2nimx)
m= 1
so tha t
\r j(±x + iy)\ = e~ ] ^ [ | l -oo
I - e -*nu>ye,
m= 1
Now| j g—2nyg2m m x | _
so tha t
^ g—2iry g2irimx _ I j g —2irj/g—27rimi I
\r)(x + iy)\ = \r } ( -x + iy)\,
as asserted. □
The fundamental transform ation formulae of r/(z) are [32, pp. 17-18], [34,
Vol. 3, p. 113]
7)(z + 1) = eni/12rj(z), r) = \ /^ iz r ) (z ) , (3.1.2)
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CHAPTER 3. KRONECKER’S THEOREM 88
where the branch o f the squareroot is taken so th a t i t has the value 1 at
z — i.
I t follows from (3.1.2) th a t i f a ,p ,y , 8 £ Z satisfy aS — P7 = 1 then
77 O /g + f ) = ev V + < ^ (*), (3.1.3)
where e = e (a ,P ,y ,8 ) and |e| = 1, see [32, Prop. 3, p. 17].
Hence
V'a z + jT ,7 z + 8 , = h 'z + ^ M z ) ! 4 (3.1.4)
Now le t ax2 + bxy + cy2 and a'x2 + b'xy + dy2 be two positive-definite,
integral, binary quadratic forms of discrim inant d (so th a t a, a' > 0, c, d >
0, d < 0) which are equivalent, so th a t there exist a ,P ,y ,5 G Z w ith aS —
Py = 1, say
a'x2 + b’xy + dy2 = a(Sx + Py) 2 + b(8 x + Py)(yx + ay) + c( 7® + a y)2.
Then
and
Set
Then
a' = aS2 + bSy + cy2
b' = 2a8P + byP + b8 a + 2cya.
b + V d
az + p _ b ' + yfd yz + 8 2 a'
G H
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CHAPTER 3. KRONECKER’S THEOREM 89
and
Thus, by (3.1.4), we have
\y z -\-8 f = —.a
4a' fb + V d \
n * J a H * j (3.1.5)
We note tha t
rj(iy) € E+, e ~ ^ - q e E + (3.1.6)
for y > 0, so th a t
V'1 + iy '
(3.1.7)
E xam p le 3.1.1. We consider the form x2 + 2y2 o f discrim inant —8. Here
a = 1,6 = 0 ,c = 2,d = —8. As x2 + 2y2 is equivalent to 2x2 + y2 (since
2x2 + y2 = (Ox + (—1 )y ) 2 + 2 ( lx + 0y2)) we have a' = 2, b' = 0, d = 1, d' =
d = — 8 . 77ms by (3.1.5) we hove
48 \
4
H 4 j = 2" { 2 )
Then by (3.1.7) we deduce
Taking logarithms o f (3.1.5) we obtain
'b + y/dlog a — 4 log V
2 a= log a' — 4 log V
V + \ fd2a!
(3.1.8)
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CHAPTER 3. KRONECKER’S THEOREM 90
In particular, as ax2 + bxy + cy2 and cx2 — bxy + ay2 are equivalent forms
(since ax2 + bxy + cy2 = c(0x + 1 y ) 2 — b( Ox + ly ) { —lx + 0y) + a(—lx + 0 y)2),
we have by (3.1.8) and Theorem 3.1.1
log a — 4 log Vb + y/d'
2 a= log c — 4 log
= log c — 4 log
V
V
'- b + Vd2c
’b + Vd2c
. (3.1.9)
We close th is section by noting the follow ing two im portant relationships
satisfied by the Dedekind eta function, which follow from the theory of theta
functions, see for example [18, p. 275]: for ^ € H
' 1 + z 'V (!) rj(2z) = em/24if(z),
+ lG r)(2zf = e ~ ^ 3rj’1 + z'
(3.1.10)
(3.1.11)
3.2 W eber’s functions
For z e H Weber’s functions f (z ) ,f i(z ) ,f2(z) are defined in terms o f the
Dedekind eta function by
f(z) = v ,
*?(§)h i? ) =rj(z) ’
f2(z) = V2rj{z) '■
(3.2.1)
(3.2.2)
(3.2.3)
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CHAPTER 3. KRONECKER’S THEOREM 91
see [34, Vol. 3, p. 114]. From (3.1.10), (3.1.11), (3.2.1), (3.2.2) and (3.2.3)
we obtain
(3.2.4)
and
/ ( * ) ' = A M 8 + / 2(* )8 (3.2.5)
see for example [34, Vol. 3, p. 114].
From (3.1.6) and (3.2.1) - (3.2.3) we see tha t i f m 6 N then f ( y /—m),
/itV^m), e R+.
E xam p le 3 .2.1. Prom Example 3.1.1 and (3.2.2) we deduce
in agreement with the value given in [34, Vol. 3, p. 721].
3.3 Kronecker’s limit formula
Throughout th is section, ax2 + bxy+cy2 is a positive-definite, integral binary
quadratic form of discrim inant d = b2 — 4ac.
We saw in Theorems 2.4.2, 2.4.3 and 2.4.4 th a t Rademacher’s theorem
allowed us to evaluate the sums
M V = 2 ) = 21' 4
(m ,n )^( 0,0)m ,n€ Z
(m ,n )^( 0,0)m,n€Z
(m,n)^(0,0)
m 2 + m n + 2n2'( - 1)
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CHAPTER 3. KRONECKER’S THEOREM 92
However i t does not seem possible to use Rademacher’s theorem to evaluate
the more general sums
“ , ( - 1) " y V ( - 1)" W ( ~ l) " +n^ am2 + bmn + cn2 ’ am2 + bmn + m 2 ’ am2 + bmn + cn21
m ,n = —oo m ,n = —oo m ,n = —oo
where the dash (') indicates th a t the term (m, n) = (0,0) is om itted. For
these we use a theorem o f Kronecker, known as Kronecker’s lim it formula,
which asserts th a t as s —* 1+ we have0°
V ' r— 5— ^ ----------j r - = • — !— + K (a ,b ,c) + 0 ( s - l ) , (3.3.1)^ (am2 + bmn + cn2)s . / } s — 1
771,71=—OO V ' V I I
see for example [32, Theorem 1, p. 14], where d — b2 — 4ac < 0 and
T.. , . 47T7 2 tclo g \d\ 27r . 87r ,
J f(o ’ 6' c) = ^ f - W + ^ log a ^ log
b + y/d^ “ 2 7
. (3.3.2)
By (3.1.9) we see th a t
K (a ,b ,c ) = K (c,b ,a ). (3.3.3)
00Theorem 3.3.1. (i) £ - i + = 2g(4a, 2i», c )-g (a , 6, c),
771,71=—OO(m ,9l)^(0,0)
771,71=—OO(171,71)^(0,0)
^ ( i ) m+n 27Tlog4 ,(m) > — ; r = ------- 7= ----- 2 A (4a, 26, c)
^ am2 + bmn 4- cn2 */OF771,71=—OO V I I
(m ,n )^(0 ,0 )
—2K(a, 26,4c) + 2Ar(a, 6, c).
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CHAPTER 3. KRONECKER’S THEOREM 93
Proof, (i) Let s > 1. Then
( i + ( - i nY ' u + i - i n = y '' (am2 + bmn + cn2)s ^ (am2 + bmn + cn2)s
n = —oo N 7 m ,n = —oo v 7m even
oo ^
= E' (4am2 + 2 bmn + cn2) 3 *m ,n = —oo v 7
Hence, as 4am2 + 2bmn + cn2 has discrim inant
(26)2 — 4(4a)c = 462 — 16ac = 4(62 — 4ac) = 4 d,
we obtain, by Kronecker’s lim it formula
(~ l)m______^ (am2 + bmn + cn2) 3m ,n =—oo 7
oo oo
= 2 V ' _________ -_____________ V '_ -________(4am2 + 2 bmn + cn2) 3 (am2 + bmn + cn2 ) 3
m ,n = —oo N 7 m ,n = —oo v 7
= 2 K (A a ,2 b ,c )-K (a ,b ,c ) + 0 ( s - l ) .
Letting s —> 1+ we obtain
00 > ( 1lim V t , >■ ;--------- r r - = 2K(4at 2bt c ) - K ( a , b t c).
s_>1+ (am2 + bmn + cn2) 3 v ’ ' y 5 ’ ’m ,n = —oo 7
°° > (—i ) mAs > t— -— --------------— converges, we obtain the asserted result.
' (am2 + bmn + cn2)m ,n = —oo v 7
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CHAPTER 3. KRONECKER’S THEOREM 94
(ii) We have by part (i) and (3.3.3)
y ( - i ) n = y ______( - i ) mam2 4 - hm.n. A- m? Q/p -]
o
- E ( - 1 ) ’
am? + bmn + cn2 ^ an2 + bnm + cm?m ,n=—co m ,n = —oo
cm2 + bmn + an2m ,n = —oo
= 2K (4c, 26, a) - K (c, 6, a)
= 2i f (a, 26,4c) — i f (a, 6, c).
(iii) Let s > 1. We have
y (1 + ( - ! ) " ) ( !+ (-!)") = y V 4(am? 4- bmn + cn2)s ^ (am? + bmn + cn2) 6
m ,n =—oo N ' m ,n = —oo ' 'mtn even
1 00 — Y 'As~l *—J (am? + bmn 4- cn2)sm ,n = —oo K 7
i
so tha t
y = f_}_ _ 4 y - ' 14— (am2 + 6mn + cn2)5 I 4s-1 / (am2 + bmn + cn2) 3m ,n = —oo ' / \ / rn,n=—oo ' 7
( - 1)«00- y ________
(am2 + bmn + cn2)s77l,n=—OO v 7uu
- E' ( - i ) "' (am2 + 6m n 4- cn2)6
,71=—OO x 7
Now
— i = g-C3- 1)1 4 _ i43~i
= (1 — (s — 1) log4 + 0 ((s — l ) 2)) — 1
= - ( s - 1) log 4 + 0 ((s - l ) 2)
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CHAPTER 3. KRONECKER’S THEOREM 95
so tha t
1 \ 00
£4 3 - I * J ^ ( a m 2 _|_ f} m n 4 . o t 2 ) 3771,71=—00 v '
(-(> - 1) log4 + 0 ((s - l)2)) ( , 1 , + /s'(a, 6, c) + 0 (s - 1)
—27rlog4>/MI
+ 0(s — 1).
Hence le tting s -+ 1+ and appealing to (i) and (ii) we obtain
(_ l)m +n — 27T log 4£ '
7711 Tl - OOam2 + 6mn + cn2 - ^ ~ W-*,2f,,c) - c))
—(2 if (a, 26,4c) — K (a, b, c))
as asserted. □
Appealing to Theorem 3.3.1 and (3.3.2), we obtain the follow ing theorem.
771,71=—OO
Theorem 3.3.2. (i) ^
m f r (-1)771,71=—C
OO
(«o £ '
( - 1)" 87ram2 + bmn + cn2
log / l'b + Vd'
2 a
87T
771,71=—OO
am2 + bmn + cn2
( _ ^ ’w+n
am2 + bmn + cn2
v f f l
87T
log
log
/2
/
'& + V 3 '2a
6 + Vd.2a
Proof, (i) By (3.3.2) we have
K (a ,b ,c ) = - B lh iM + - - ^ l o gs /R vW v13f v/RI
Vb + -\/d
2 a
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CHAPTER 3. KRONECKER’S THEOREM 96
and
K(4a, 2b, c) =47T7
~ 7W \27T7
27r log 4 |c?| 27rlog4a 87T
v / iR V * Rlog V
'2 b + y/4d 8 a
7rlog4 7rlog |d| 7rlog4 7rloga
V R [ y jd f v13f \ /P i y p f47T
log Vb + Vd
4 a
27T7 7T log |d| 7T log a 47rlog
b + Vd4 a
so tha t
2 i^ (4a, 2b, c) — i f (a, b, c)
47T7 27r log |d| 27r log a 87r
a /R V P i -Mlog V
b + Vd4 a
47T7 2n log \d\ 27rloga 87r ,H 7 = ----------------- 7 = = - 7 = log
87T
W i
■ M VW\^b±& '\
VW \ V W \V
b + Vd2 a
logV \ 2cT)
( b+Vd\{ 4a )
Then, appealing to Theorem 3.3.1 (i) and (3.2.2), we obtain
£ ( - i ) ’772,73,=—OO
(m ,n)^(0,0)
am2 + bmn + cn28?r
V W \
87T
~ W \
logV (*8?)V
log
( b+\/d\V 4a ;
b + Vdf i 2 a
(ii) By (3.3.2) we have
47T7K (a , b, c) —
27rlog|d| 27rloga 87r , —7= = ---------------- + ------; = --------- 7= log 77v ld f v ld i v W V H
'6 + V d '2a
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CHAPTER 3. KRONECKER’S THEOREM 97
and
K (a , 2b, 4c)47T7 27T log 4|d| ^ 27rloga 8 tt
vipf , / mlog
27T7 7rlog4 7rlog\d\ ^ 7rloga 47r
v f f l v W vTdf v M >/PT
V
log
2 6 + \/4 d ' 2 a
b+ VdV '
so tha t
2 K (a , 2b, 4c) — K (a , b, c)
47T7 27rlog4 27rlog|d| 27rloga 87r+
% / H v W % / H v ' M l v f f llog V
'b + Vd
47T7 ^ 27T log |d| 27r log a ^ 871" ^
v W y p f v r i y R
47r log 2 87r
V'b + Vd'
2 a
log■n { < ¥ )
V (I 2a )
Then, appealing to Theorem 3.3.1(ii) and to (3.2.3), we obtain
( - 1)
m .n —~ooam? + bmn + cn2
4 7 r , _ o 8 ? r I
\ m v w \
87r , „ ( b + Vd
V' b + \/d \ < a )
Vf in -v tAI 2a )
log f i2 as/¥\
(iii) Prom parts (i) and (ii) we have
-2tf(4 a , 26, c) - 2 t f ( a ,2b, 4c) + 2K (a , b, c)
= —(2i^(4a, 26, c) — K (a , b, c)) — (2K (a , 2b, 4c) — K (a , b, c))
87r
V W \log
Vb ± V d \
4 a )
47t log 2 8 n
VW \ V\d\log
( te a )
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CHAPTER 3. KRONECKER’S THEOREM 98
= 27T log 4 8tr I7? ( ^ ) 117? C +4 ^ )
V P f "s/Pi
Appealing to Theorem 3.3.1(iii), and to (3.2.2), (3.2.3), and (3.2.4), we
obtain
£771,71=—OO
(m ,n)^(0,0)
am2 4- bmn + cn287T
7 R
87T
log 2 2
log
Ab + Vd
2 a
V 2
AW V d '
2a
- l
87T
V\d\log / b + V ]d [
2a
completing the proof. □
In the next theorem we reprove Theorem 2.4.3 using Theorem 3.3.2.
T heo rem 3.3.3.
f * ( - l ) m2—* m 2 + 2n2
771,71=—OO(m ,n)#(0,0)
V 2 l0g2 '
Proof. From Example 3.2.1 we have
h (v^2) = 21/<4.
Taking a = 1, b = 0, c = 2, d = b2 — 4ac = —8 in Theorem 3.3.2 (i) we obtain
t771,71=— OO V
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CHAPTER 3. KRONECKER’S THEOREM 99
- - t H 21/4>
~ 7 I los2as required. □
Taking a = 1, b = 0, c = A e N, so tha t d = b2 — 4ac = —4A < 0, in
Theorem 3.3.2 we recover a result of Zhang and W illiam s [37, Theorem 3, p.
189] proved in 1999. We give the results in terms of Ramanujan’s functions
3k = 2 -1' 4/i(v /=A), Gj = 2 -1'7 (v '= A ),
see [29, p. 27].oo / ' i \m
T h e o rem 3.3.4. (i) £ log (2 *}) .771,71=—OO v
(m ,n)^(0,0)
( ~ l ) n 7T
771,71=—OO(m ,n)^(0,0)
(“) E rf+w = _ log(2G*)771,71=—OO v
(m ,n)^(0,0)
Proof, (i) By Theorem 3.3.2(i) we have
(-1)™ _ 8tt
rm ,n = —ooE' O T -
4?r log/i(V^A)VA
47f log(21/45A)Va
* log(2).V a
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CHAPTER 3. KRONECKER’S THEOREM 100
(ii) By Theorem 3.3.2(ii) and (3.2.4) we have
'sAJ
£ 'm ,n = —oo
( - i ) " m 2 + An2
47T7x47T
logV a & / ( vC A )/i (x/= A )
log (2 -V V ( v^ A )2 -1',V i (vc A ))
^ lo g f e G i )
= ^ l o g ^ G j )
(iii) By Theorem 3.3.2(ii) we have
£ 'm ,n =—oo m2 + An2
8?r
2\/A 47T^ lo g / ( v ^ X )
^ l o g ^ C * )
^ lo g ( 2 G y ,
completing the proof. □
The firs t four values in Table V I of [34, Vol. 3, p. 721] are
= 21'4,/ , (V = 2 ) = 2*/*
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CHAPTER 3. KRO NECKER’S THEOREM
/(> /= 3 ) = 21/3,
f i (V —4) = 23//8 (w ith a typo corrected).
From (3.2.4) and (3.2.5) we deduce
A ( 7 = 1 ) = h ( 7 = 1 ) = 2 V 8
/ (7=2) = 21'8 (72 + l ) 1/8, A (7=2) = 21'8 (75 - l) V8,
A (7=3) = 21'18 (2 + 7s)1/8 A (7=3) = 2^12 (2 - 73 ) 1/8
/ (7=4) = 2‘/« (1 + 72)1/4 A (7=4) = 2^le ( l - 72) V4
HenceSi = 2-V8, G l = 1,
\ 1/892 1, G2 = 2- 1/8 ( ^ + l ) ,
93 = 2-1/6 (2 + ^ ) 1/8, G3 = 21//12,
f t = 2^ , G4 = 2-3/ 16 ( l + V ^ )1/4.
Appealing to Theorem 3.3.4 we obtain the follow ing series evaluations.
T heo rem 3.3.5.
( - l ) m _ 7TOO
E - ~ > * 2’m 2 + n 2 v.771,7l<— OO
(m ,n)#(0,0)
^ ( — l ) n 7T ,^ m2 + n 2 ~ ~ 2m,n=~oo
(m ,n)^(0,0)
00 / 1 \m + n
2 1 2 = — 7r l ° g 2 -td ." -1- 77"m 2 + n2771,71=—OO
(m ,n)#(0,0)
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CHAPTER 3. KRONECKER’S THEOREM
Theorem 3.3.6.
” J _ 1 )m _ _ L l0o;2^ m? + 2 n 2 V2
771,n = —oo v(m ,n)^(0,0)
( “ I ) ” _ ^ l n ^^ m 2 + 2n2 “ ~ 2 \ / l 2 \ / Im ,n =—oo v v
(m ,n)^(0,0)
“ (_!)■»*■ _ ____ j _ „
J iz L * , ™2 + 2" 2 2^ 2 2V2(m ,n)^(0,0)
Theorem 3.3.7.
(—l ) m _ 7T 0 7r
771,71=—OO v v(m ,n)#(0,0)
Y '' ( ~ l) n ^ ^ i
m2 + 3»2 _ " S v ? g + 2VS771,71=—OO v v
(m ,n )/(0 ,0 )
~ ( _ ! ) m + n 4?r
^ m 2 + 3n2 “ 3v/3771,71=—OO V
(ro,n)^(0,0)
Theorem 3.3.8.
_ ( - ! )"» _ 3 f i j^ m 2 + An2 4 g ’
771,71=—OO(m ,n)^(0,0)
E ( — l ) n 7T 7T
m 2 + 4n2 ~ ~ 8 ® 2 ®771,71=—OO
(m ,n)^(0,0)
« ( _ ! ) « + » 7T n 7T ,
m 2 + 4n2 “ 8 g 2 S771,71=—OO
(m ,n)^(0,0)
log (\/2 + 1)
log (\/2 + 1)
5g (2 + V 3 ) ,
5g (2 + \/§ ) ,
I + V 2) ,
i + V2).
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CHAPTER 3. KRONECKER’S THEOREM 103
3.4 Final results
The evaluation o f Weber’s functions \f(z )\, \f i(z )\ and 1/ 2(2)! at quadratic
irra tiona lities z — (d < 0) has recently been carried out by Muzaffar
and W illiam s [26]. P u tting the ir evaluation together w ith Theorem 3.3.2, we
obtain the evaluation o f the in fin ite series
y V (■-1)™ y V ( -1 ) " y V (-1)™ +"^ am2 + bmn + cn2 ’ am2 + bmn + cn2 ’ ^ am2 + bmn + cn2 ’
m ,n = “ 0 0 771,71=—oo 771,71= —oo
for a positive-definite, prim itive, integral binary quadratic form ax2+ 6 xy+ q /2
of discrim inant d (< 0).
In order to state the result of Muzaffar and W illiam s it is necessary to
introduce some notation. Let d be the discrim inant of a positive-definite,
prim itive , integral b inary quadratic form ax2+bxy+cy2 so th a t d = b2 —4ac =
0 or 1 (mod 4), a > 0, c > 0, d < 0. The conductor / o f d is the largest
integer such th a t d / f 2 = 0 or 1 (mod 4). We set A = d / f 2. The set
o f classes of positive-definite, p rim itive , integral, binary quadratic forms of
discrim inant d(< 0) under the action of the modular group
r ,s , t ,u € Z, ru — st = 11
is denoted by H (d). I t is well known th a t H (d) is a fin ite abelian group w ith
respect to Gaussian composition, see for example [9]. The group H (d) is
called the form class group. The order of H (d) is the class number h(d). We
w rite [a, 6, c] for the class containing the form ax2 + bxy + cy2. The iden tity
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CHAPTER 3. KRONECKER’S THEOREM 104
of H (d ) is the class
1 =
1,0, -d
1, 1,1 - d
i f d = 0 (mod 4),
, i f d = 1 (mod 4).
The inverse o f the class K — [a, 6, c] E H (d) is the class K 1
H (d). Let A i , . . . A s be generators of H (d) chosen so th a t
[a, - b, c] €
h i/i2 ■■•hs = h(d), 1 < h i | h2 \ • • • | hs, o rd ^ * ) = h i( i = 1 , . . . , s).
For K = A * 1 • • • A kss E i/(d ) and L = A [x ■ ■ ■ Ae/ e H (d) we define
X ( K , L ) = e2ni( ^ + - +h^ ) .
I f p is a prim e w ith ^ = 1, we le t x \ and rc2 be the two solutions of x2 = d
(mod 4p), 0 < x < 2p, w ith x \ < x2. We define the class K p E H {d) by
x \ — d
so tha t
K p = P> x i ’ 4p
’ 4p
For K ( j t I ) E -ff(d) the Bernays constant [7, Teil I, §3, §4, pp. 36-68] is
defined by
pa h '
I t is known th a t j (K ,d ) is a nonzero real number such th a t j (K ,d ) =
j ( K _1, d) [26, Lemma 7.6]. For K E H {d) and n E N we define H x{n )
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CHAPTER 3. KRONECKER’S THEOREM 105
to be the number of integers h satisfying
h2 - d0 < h < 2 n, h2 = d (mod 4n),
Further for n 6 N and K E H (d) we set
n, h,4 n
= K .
YK ( n ) = Y , x ( K , L ) H L (n).L£H{d)
For K E H (d) and a prim e p we set
A (K ,d ,P ) = Y ^ 1 -d=o F
Then we define for K E H(d)
l ( K , d ) = n ( l + x(-K ' K p ) ) l [ A ( K , d , p ) .p > pi?
r f/
We also set
*«> - n (* - £ ) •(f)=i
Finally for K E H(d) we define
TTi/jy- _ * V \ d \ \ ' / r T S - \— 1 ^ l ( ^ ) P ( T J \
48W r ^ * {L ’ K ) J iL id jL±I
We are now ready to state the result o f Muzaffar and W illiam s [26, Theorem
2]. I t is convenient to w rite fo(z) for f(z ) . The power of 2 occurring in the
nonzero rational number r is denoted by v2 (r), so th a t
j,2(24) = u2 (23 • 3) = 3,
1/2 ( t ) = U2^ 2 ' 3-1 ’ =2’
= ^ ( 2' 3 ■ 3-1 ■ 7) = - 3-
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CHAPTER 3. KRONECKER’S THEOREM
T heorem 3.4.1. Let K = [a,b,c] € H {d). Set
q0 = a + b + c, q\ = c, q2 = a,
1, i fq i = 2 (mod 4),
1, i f q% = 0 (mod 4), b = 1 (mod 2)
1/2, i f q% = 0 (mod 4), b = 0 (mod 2)
2, i f qi = l (mod 2),
fo r i = 0 ,1,2,
M 0 =
M x =
M 2 =
2a Aq, Ao(2a + 6), ~ ( a + 6 + c)
2aAi, Ai6, y c
A2a, A26, 2A2c
€ # (A^d),
m, = 2 - 21 _ " 2 ( A i ) = <
0, * / Ai = 1,
1, i f Xi = 2,
-2 , i f Xi = 1/2,
/o r i = 0,1,2. Then
b + \ /d/*
fo r i — 0,1,2.
2a =(i:1/4
Prom Theorems 3.3.2 and 3.4.1 we obtain the following theorem.
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CHAPTER 3. KRONECKER’S THEOREM 107
T heo rem 3.4.2. With the notation o f Theorem 3.4.1,
(i) t am? + bmn + cn2771,71=—OO
^ {i log g ) +mi (gi|))2- ,( /) I o g 2 + E ( K , $ _ E ( M u A?(j)
uu
(M) E ( - i ) "am2 + bmn + cn?
771,71=—OO
TPf {1los © + to*2+ A 2 d )
on) E00 / . \ m -|_n(-1 )*
am2 + bmn + cn2771,71=—OO
. 1,0g g ) +mo(g||))2-W/,log2 + i5(if,d) _ £(Mo,^)
We conclude th is thesis w ith an example illus tra ting Theorem 3.4.2. We
take
a = 1, 6 = 0, c = 19
so tha t
d = -7 6 , / = 2, A = -1 9 , g ) = -1 , M f ) = 1.
K = [1,0,19] e H ( - 7 6 ) ,
qQ = 20, qi = 19, q2 = 1,
A° = - , Ai = 2, A2 = 2,
Mo = [1,1,5] € # ( - 1 9 ) ,
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CHAPTER 3. KRONECKER’S THEOREM 108
m 0 = 2 — 21“ V2(Ao) = 2 - 4 = -2 .
Muzaffar and W illiam s [26] have shown tha t
E (K , d) = log ( J jL ) , £ (M „, -1 9 ) = 0,
where 0 is the unique real root of x3—2x—2 = 0. Thus, by Theorem 3.4.2(iii),
we have
(_ i)m +n 8tt / l . , - 2 1 . _ . ( 6 \ \
^ m 2 + 1 9 n 2 “ i/7 6 \4 2 ' 3 ' 8 S + g (,21/3 J J47T ( \ 1 1 \
= - 7 f 5 ( 2 lo« 2 - 6 l0s2 + l0 g 9 - 3 1Og2)
m ,n =—oo
47Tlog 0.
We have proved
T h eo rem 3.4.3.
V - ' ( - l ) m+n 47T , „^ m 2 + 19n2m ,n = —oo
S im ilarly we can show from Theorem 3.4.2(i), tha t
V ' 4 ^ = — ^= lo g (2 7 + 5\/29),^ m 2 + 58n2 y/58m ,n = —oo v
a result given by Zucker and Robertson in [38].
We finish by remarking th a t the series in Theorem 3.4.2 are related to
series o f the form V ------------= ------, k e N , see [35], [36], [37].n sinh(\/& 7m) 1 J 1 J
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Conclusion
We conclude th is thesis by mentioning a possible line of fu ture research. We
have noted th a t certain in fin ite series evaluations such as
f ; ± v r _ _ j i _ , o g 2 _ ^ L log('2 + V 3 ) i^ m 2 + 3n2 3\/3 2 \/3 V /m ,n = —oo v v
(m,n)^(0,0)
mUL2 = _3 ^ l0g2+2 ^ l0g(2 + V )’m ,n = —oo v v(m ,n )^ (0 ,0 )
“ ( _ l ) m + n ^ 4?T
^ m 2 + 3n2 “ 3 ^3m ,n=—oo v(m ,n )# (0 ,0 )
see Theorem 3.3.7, follow from Kronecker’s lim it form ula but do not appear to
be capable of deduction from Rademacher’s theorem. I t is therefore natural
to conjecture th a t there is a generalization of Rademacher’s theorem which
gives the above results as special cases. Much remains to be investigated.
109
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Bibliography
[1] A .W . Addison, A series representation fo r Euler’s constant, Amer.
M ath. M onth ly 74 (1967), 823-824.
[2] S. Alaca and K . S. W illiam s, Introductory Algebraic Number Theory,
Cambridge U niversity Press, 2004.
[3] T . M . Apostol, In troduction to Analytic Number Theory, Springer-
Verlag, 1976.
[4] T . M. Apostol, M athem atical Analysis, Addison-Wesley, Reading, Mas
sachusetts, 1965.
[5] E. Asplund and L. Bungart, A F irs t Course in Integration, H olt, Rine
hart and W inston, 1966.
[6] F. L. Bauer, Eine Bermerkung zu Koechers Reihen f i i r die Eulersche
Konstante, Sitz. Bayer. Akad. Wiss. M ath.-N atur. K l. (1990), 27-33.
[7] P. Bernays, Uber die Darstellung von positiven, ganzen Zahlen durch
die prim itiven, bindren quadratischen Formen einer nicht-quadratischen
Discriminante, Ph. D. thesis, Gottingen, 1912.
110
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
BIBLIOGRAPHY 111
[8] B. C. Berndt and D. C. Bowman, Ramanujan’s short unpublished
manuscript on integrals and series related to Euler’s constant, CMS
Conf. Proc. 27 (2000), 19-27.
[9] D. A. Buell, B inary Quadratic Forms, Springer-Verlag, New York, 1989.
[10] H. Cohn, Advanced Number Theory, Dover Publications, New York,
1980.
[11] L. E. Dickson, Introduction to the Theory of Numbers, Dover, New
York, 1957.
[12] K . Dilcher, Generalized Euler constants fo r arithmetical progressions,
M ath. Comp. 59 (1992), 259-282.
[13] I. Gerst, Some series fo r Euler’s constant, Amer. M ath. M onthly 76
(1969), 273-275.
[14] J. W . L. Glaisher, On Dr. Vacca’s series fo r 7 , Quart. J. Pure Appl.
M ath. 41 (1909-10), 365-368.
[15] E. R. Hansen, A table of series and products, Prentice-Hall, Englewood
C liffs, NJ. 1975.
[16] K . Hardy and K . S. W illiam s, The Green Book of M athem atical Prob
lems, Dover, NY, 1997.
[17] L .-K . Hua, In troduction to Number Theory, Springer-Verlag, Berlin Hei
delberg New York, 1982.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
BIBLIOGRAPHY 112
[18] J. G. Huard, P. Kaplan and K. S. W illiam s, The Chowla-Selberg formula
fo r genera, Acta A rith . 73 (1995), 271-301.
[19] E. Jacobsthal, Uber die Eulersche Konstante, K . Norske V id . Selsk.
S krifter (Trondheim) (1967).
[20] F. John, Identitaten zwischen dem Integral einer willkurlichen Funktion
und unendlichen Reiben, M ath. Ann. 110 (1934), 718-721.
[21] M. Koecher, Einige Bermerkungen zur Eulerschen Konstanten, Sitz.
Bayer Akad. Wiss. M ath.-N atur. K l. (1990), 9-15.
[22] D. H. Lehmer, Euler constants fo r arithmetical progressions, Acta A rith .
27 (1975), 125-142.
[23] J. Lesko, Sums o f harmonic-type series, College M ath. J. 35 (2004),
171-182.
[24] J. J. Y. Liang and J. Todd, The Stieltjes constants, J. Res. Nat. Bur.
Standards Sect. B. 76 (1972), 161-178.
[25] M. S. Longuet-Higgins, Shooting fo r n : the bowstring lemma, M ath.
Gaz. 84 (2000), 216-222.
[26] H. Muzaffar and K. S. W illiam s, Evaluation o f Weber’s functions at
quadratic irrationalities, JP Jour. Algebra Number Theory Appl. 4
(2004), 209-259.
[27] H. Rademacher, Some remarks on F. John’s identity, Amer. J. M ath.
58 (1936), 169-176.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
BIBLIOGRAPHY 113
[28] S. Ramanujan, The Lost Notebook and Other Unpublished Papers,
Narosa, New Delhi, 1988.
[29] S. Ramanujan, Modular equations and approximations to 7r, Q uart. J.
M ath. 45 (1914), 350-372. (Collected Papers of S. Ramanujan, Chelsea
Pub. Co., New York, N .Y., 1962, pp. 23-29.)
[30] W . Rudin, Principles o f M athem atical Analysis, 2nd edition, McGraw-
H ill, 1964.
[31] H. F. Sandham, Advanced Problem No. 4353; Solution by D.R. Barrow ,
Amer. M ath. M onthly 58 (1951), 116-117.
[32] C. L. Siegel, Advanced A na lytic Number Theory, Tata In s titu te of Fun
damental Research, Bombay, 1960.
[33] G. Vacca, A new series fo r the Euleiian constant j = 0 .577..., Quart.
J. Pure Appl. M ath. 41 (1909-10), 363-364.
[34] H. Weber, Lehrbuch der Algebra, Vols. 1-3, republished Chelsea, N .Y,
1961.
[35] K . S. W illiam s, Sums o f two squares and the series V ——~r------, part* m smh nm
771=1
I, Carleton Coordinates, Carleton University, Ottawa, December 1993,
pp. 4-6.
[36] K. S. Williams, Sums of two squares and the series ) — . , part' m smh 7rm
771=1
I I, Carleton Coordinates, Carleton University, Ottawa, May 1994, pp.
4-5.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
BIBLIOGRAPHY 114
( _ l ) m—7:— and related series, m + A n*m,n= —00
(m ,n)^(0,0)in Mathematical Analysis and Applications, edited by Th. M. Rassias,
Harmonic Press, Inc., Florida, 1999, pp. 183-196.
[38] I. J. Zucker and M. M. Robertson, A systematic approach to the evalua
tion o f E (am ?+ bmn + cn2) s, J. Phys. A. M ath. Gen. 9 (1976),(m ,n)^(0,0)
1215-1225.
OO[37] N .-Y. Zhang and K . S. W illiam s, V
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