thelattice points of a circle.rspa.royalsocietypublishing.org/content/royprsa/106/739/...0 — n —...

478

The Lattice Points of a Circle.By J. E. Littlewood, F.R.S., and A. Walfisz.

(With a Note by Prof. E. Landau.)

(Received April 22, 1924.)

1. Let r ( x)denote the number of ways in which the positive integer x cai be expressed as the sum of two squares (positive, negative or zero), and let

R(x) = S r( S 1.0 — n — X 0 ~ p 2+q2~ x

Thus R {x) is the number of “ lattice-points ” (points whose co-ordinate: p, q are integers, positive, negative or zero) in or on the boundary of tht circle with centre at the origin and radius y/ I t is trivial that

(1.1) R (x) — nx = 0 (x*

it has been shown by Hardy* and Landauj that the relation

R (x) — nx — (xa)

is not true for any constant a < and it has been known for some time tha

R (x) — nx = 0( :*).

A very important advance is due to van der Corput, who proved in a recent paperj that

(1.2) R (x) — nx — O (x&+e)where 0 is some constant less than -J. For the corresponding problem concerning

D (x) = S (l ( ),

where d (n) is the number of divisors of ft, van der Corput obtains§ the more precise result

(1-3) D ( z ) - D 0 (z) = O (*»+'),

* G. H. Hardy, “ On the expression of a number as the sum of two squares, ! Quarterly Journal of Math., vol. 46, pp. 263-283 (1915).

t E. Landau, “ Uber die Gitterpunkte in einem Kreise (II),” Oottinger Nachrichkn, pp. 161-171 (1915).

J J. G. van der Corput, “ Neue zahlentheoretisehe Abschatzungen,” Math. vol. 89, pp. 215-254 (1923).

§ J. G. van der Corput, “ VTerscharfung der Abschatzung beim Teilerproblem," Math. Annalen, vol. 87, pp. 39-65 (1922). A number of misprints in this paper are corrected by the author in the errata attached to vol. 89.

on June 4, 2018http://rspa.royalsocietypublishing.org/Downloaded from

http://rspa.royalsocietypublishing.org/

The Lattice Points o f a Circle. 479

here D0 (x) is a certain elem entary function whose leading term is x log x, is an arb itrary positive number, and (H) is some constant less than 3^5-.The present writers, applying a m ethod used by H ardy and Littlewood* to

rove th a t

1.4)

iscovexed, independently, th a t it is possible to prove (1.2) with

1.5) 0 = ttV.

The result (1.5) is new, b u t we do not regard this as the main point of he present paper. Doubtless no proof of (1.2) (with 0 < | ) can be altogether a s y ; b u t whereas van der Corput’s m ethod is probably the m ost formidable rgum ent in the whole of pure m athem atics, it will be found th a t the proof liven here is both reasonably short and not impossible to grasp in its ntirety .

Our proof rests on the possibility of finding an expression as a (finite or nfinite) series for R (x) — nx, sufficiently approxim ate in itself, and amenable in essence) to the argum ent of §§ 5-7 below. There are, in point of fact,, ilternatives to the expression we adopt (which is based on a result due to iVigert); the several versions of the proof all lead, however, to the same falue of 0 .

2. We suppose always x > 2, and write for convenience s = Thesymbols cx,c2, ..., c30 denote positive constants depending only on a, a positive constant introduced in § 5, and 0 ’s depend only on e, and on a when that is a param eter.

Lemma 1. We have

R (x) — nx = i x* 2 -f- [x^L*'),n == 1

where.1 __ i_

(2.1) an = e~u uis cos (271 — | du.Jo

This result is due to W igert,f who proves it, indeed, uniformly in > 1 and with a remainder term

o j - ! + 0 ( 4 \ S J

* See E. Landau, “ Uber die £-Funktion und die L-Funktionen,” Math. Zeitschr., vol. 20, pp. 105-125 (1924).

t S. Wigert, “ Uber das Problem der Gitterpunkte in einem Kreise”, Math. Zeitschr., vol. 5, pp. 310-318 (1919).

2 m 2

480 J. E. Little wood and A. Walfisz.

W igert’s method, an application of the Euler-Maclaurin sum-formula, involves rather elaborate calculations, and we have thought it worth while to indicate an alternative proof, based on the double Fourier formula

£ f (p , ? ) = £ [ [ eM{mu+nv)f{ u , v) dv.p } q = — x wt, n = —‘o c j —qoJ —x

This formula is easily seen to be valid when/ „ 2 | ,,2 s

f(P> ?) = eXP

and we have

V, ?

» y y f /u2 4- v2\s 1 ~= £ cos 27 tpucos 27 exp ■< — ( --!— j f

Pl q = — o o j — o o j — oo \

— \x X e~RSdR cos (27tp v 7»R cos $) cos (2 \/& R sin </>) d<f>P,g = - 00 Jo Jo

p 30 00 00 __________

= Jtx e~RSdR + nx2 r (n) e~RSJ (2 dR*Jo n = 1 Jo

= 3rar ( i + t ) + s v / * £i V W Jo

e RS R s ® J x (2tt \/w xR) dR f

(2.2) = Tree + v 7* 2 ^-7^ I e~Mw2s J x (2tt \/wa; w2s) - f 0 (a;112) • ft = 1 V ^ J 0

On the other hand we have, in virtue of r (n) (ne),

R (a?) — 2 exp —pt q= - co

P2 -(- \S

— 2 r (n)*| 1 — exp — ( - )0 — n^.x L L \x / -

— 0 ue ) du) 4- o ( j* ue exp

- S r ( » )exp [ - ( ? ) ]n > z

u \sx! -j

du j 4- 0 (xe)

= 0 ( - — ) = 0 (xA“»+‘).\ s

Hence, from (2.2),

(2.3) R (x) — nx = x 2 - 7 J f e~u u2sJ 1 \ / n x u2s) du -j-0 (aTi l -+‘)-n - 1 v w j 0

* See G. N. Watson, “ Bessel Functions ”, p. 21, § 2.21, formula 1. t We integrate by parts, and use Watson, p. 46, §3.2, formula 5.

Since

Ji (y) — \ / (— ) cos (y — In) v ny

■less than an absolute constant for y>• 0, it follows from (2.3) that

1 00 rin) f 00 — ___ JLR (x) — nx = - xi2 - V e~uu is cos (2tt \ / wx w2* — ^

n = 1 W4 J o

+ 0 ( x - i ) 2 e~u u ~ id u - \ - 0 (x&*+e)n = 1 Jo

= - X* 2 «» + 0 ( x ^ +e),n = \ n*

ie desired result.3. Lemma 2. We have

R (x) - nx = - x* 2 ^ an + 0 ( x ^ +t).n » s * ^ +e n*

For each integer p and large £ (or s) we have, writing u2s for u in (2.1) ad integrating 2p times by parts,

2sj cos (27r f n x u — | n) (^ ^ e “2s w2s~»"j- du | .' ttn' (4tt2wx)p

ategrating once more by parts and observing th a t

™ \ ! d \2p+1da

2p + l f®du < 2 Ksm e~u2t (w2s+Ks + u2s- K) du

m=0 J 0< 2 K s ms~1 < Ks2°,

re conclude th a t, for a suitable a? (p) > 2 and x > x (p),

»i < k (—y+!= k i—Y™,'XTe \ Pnx' n

vhere K depends only on p. From this it follows th a t

- x 1 2 r (n) in\ = 0 (K x i+w<p+4)) 271 n<n>xU+t

md this by a suitable choice of p = p (e) (sufficiently large) is 0 (xT +e). Lemma 2 now follows from Lemma 1.

4. Lemma 3. We have

(4.1) R(x) — 7ix = — x1 f e~u u4s 2 cos (z — \n ) -j-71 J *-‘ ni

where

Z = 271 y / x U 2*-

For

I xi r(n)71 \ n+€ U4r

(4.2) = 0 (xi)

j* + | ) e ~ u uit cos ( V7 w2s — |

(4.2) = 0 (a*) 2 (I* du + [ (^) 2 aT1 = (x^+%

and the result follows from (4.2) and Lemma 2.5. Lemma 4. For -g-f -j- e< f , and uniformly for x~l < u < x, we ^aue

L(--) gU O (ajTTa+e).(5.1) 2W+

The main theorem follows a t once when Lemma 4 is proved. For then the sum 2 in (4.1) is 0 (xT +e), and we have

R ( « ) — nx = 0(a?l+da+e) fJz

±e~u w2* du

We prove first Lemma 5.* .

Q and Q' being integers. Then

= 0 (aA^+e) I e~w(1 u) ( x ^ +e).

Lem m aS.* Leta>0, 0 < t < L —— < Q < Q' < 0 J ; for if x ^ L

^ 2 x Ts2a;1 r + A + a.Q'V

2 = QWe write

v = [»4t- to]} n _ Q' - Q Vn — Q + nv>

n being a non-negative integer. Clearly v s l . We may suppose N ^ since otherwise

Q'2

2 = QV ------ - + 1 < 3v < 3x '9“t + A +

And we may further suppose x> c2> 2, where c2 has the property

3v 3v --< __.vn Q

* It is here that we employ the method of Hardy and Littlewood referred to in § 1.

ov ov / >— (x±C2),

The Lattice Points o f a .

: idently valid if c2 is large enough. Since

% - i ~h v “ vn > Q “h i — — 1) v = Q ' — v,

e have

483

\ v

1.3)

Let now

Q' N — 1 v — 12 e*'z/(p->+2-) _ 2 2 e« y{p2+(f,+m)2}

q = Q m = 0 m = 0< v ^ # i9°r **" + a.

\ / { / + (v» + y f ] = v7(p 2 + O {p (y) -+ y5/(y )

here the square root is taken positive for > 0,

p (y) = pn (y) = ay* + + «2?/2 + +

i a real polynomial of the fourth degree in y, and / (y) is regular a t 0. 1he number a is given by

„ _ P 2 ( L y — p 2)5.4)

iVe write further

2\4 *8 ( / + t’n2)

y (y) = s y , / .4 = 0

so that in the neighbourhood of 0

, W = « p [ * ^ + O { ( ] ) ^ ± | + (}) W +

fj}]Since z < c3 y /x ,it follows th a t

[

^ exp

(p + ^«

I y / i \ / y24-2nny"h fc=5W ' / + ^

„ , | h | J V5v n v 5v w3 “ /'3vvn\kS I #4 : V - exP ¥ 1 5 + 7 “+ ^ 2= 0 L L v n _ 5 \

v5[ V * £

^CC.

Hence

2 g*2 V{p2+ (t>»+m)2}m = 0

2 g’2 y(p- + w„J)P(m) £ y AraA m = 0 h = 0

4 = 0

V—12 g<2 s/(p -+ ^ 2)P (m )m 4

m = 0

- 2 | | v* Max/* = 0 0 < vf < I'—1

v — 1v </(!*+*.*) P (»)

m — v'

(5.5) ; c6 MaxV— 12 g « y (p '+ e .2)P(w)

m = v

484 J. E. Littlewood and A. Walfisz.

6. Lemma 6. Suppose b integral and6 > 2 ;

P (y) = + (W*-1 + ••• + p6

a polynomial with real coefficients ; <J> real, v2 integral, v2 integral and v2 - l.

Thenob-l

v'£ 2 gii/iP (m)m = Vi +1

- (2v2)2' ~~6 s Min {v2, I cosec (£ <j; (306 ! m1... m ^ ) | } *!»*,|+ ... + |w &_1l

This result is due to W eyl.f In Lemma 6 we now take

6 — 4, Vx = v' — 1, V2 — V V, 4I== (p “H Po =

We obtain, writing for brevity

(6.1) K — 12a2 \ / -j“ 2) ,

y;1 e’2 y(p-+v,-) P (w)

m = v1: c7 (v — v')» j 2 Min (v — v', | cosec | ) 1

Maxv— 1

2 gtz n/( p2+ Vn2) P (rn)

m — v'

I'm, +|.w.2| + |w3|

< c7v® { 2 Min (v, | cosec | ) 5Llw1|+|m2| + |w3|£i'

; c8v* -("v3 -j- 2 Min (v, |cosecfn |) j-8m}, m2i wi3 = l -J

v* + VM 2 Min (v, | cosec tn | )1»8

From this and (5.5) it follows th a tN - l v—12 2 eiz J{p2+ (v'‘+mn

1% = 0 m = 0

(6 .2) < c9

— c9

Nv* -f- N M

N —1 f v , "] i"Nv? + s i 2 Min (v, | cosec | )

n = 0 ^ mu m*, m3 = l ^v N —1 n 1 _2 2 Min (v, | cosec | ) f-8

mu m*2i m3 = l n = 0 * J

by the generalised Schwarz inequality. By the hypotheses of Lemma 5,

Nv* < c10x ^ r + n M s c10a^°r + ®3°.

* The “ Min ” denotes if the cosecant is infinite.f H. Weyl, “ Uber ein Problem aus dem Gebiet der Diophantischen Approximationen,

Gottinger Nachrichten (1914), pp. 234-244 (241-243). See also the same author, “ Uber die Gleichverteilung von Zahlen mod. Eins,” Math. Annalen, vol. 77, pp. 313-352 (328 -331) (1916), and again “ Zur Abschatzung von (1 + ip P Math. vol. 10,pp. 88-100 (90), (1916). The form we adopt, as sufficient for our purpose, is taken from a paper by E. Landau, “ Zum Waringschen Problem,” Math. Zeitschr., vol. 12, pp. 219-247 (224-225) (1922).

485The Lattice Points of a Circle.

lence it is easily verified, from (5.3) and (6.2), th a t Lemma 5 follows when ve have shown th a t

v N —16.3) 2 2 Min (v, |eosec tn \) < cn (x1 4"

Wj, m2> ms = l n = 0

Now from (5.4) and (6.1) we have

*» =

Sinced4y2 — f_ 5y (3y2 — 4;y2)

dy ( / + y 2) ( / + y 2)*md

3 / < 3Q2 < 3vn2, %_x < % < Q - f ^ v = Q' < 2xT, 2 > c12 ^/a?,

we have for 0 s n < N — 2

tn —tn+1 ± (vn+i — vn) izm 1m2m3p2 Min ^ — V — ®«+1 ( p + y >

> c13vv/ xnixm^m^2 L*-

(6.4) ^ ch mxni2mzip 2‘x ^ ~ ^ T>

Let now £„ (y integral) denote the interval

< V =£ {9 + !)•

We consider the part sum

riy = 2^ Min (v, | cosec | ) = 2

where 2 S is taken over those values of n between 0 and N — 1 inclusive for which tnlies in £g. Let g*be \g or ^ (g-f-1), whichever is an integer, andthe rank of th a t term of y\g for which | tn. — j is a minimum. For n we have w« < v ; for other values of n, u a < | cosec | , and it follows from(6.4) that

(6.5) % ^ v - f |cosec *«! - v + 2 ' |cosec (cl4)w1m2m3p 2x^_ %s"rk) '•,n # w* Tc

where 2 ' is taken only over Jc for which

clim1m2m3p 2 ~ 'T k < |.t .

From this it follows that

(6.6) ^ V + C15X * T~* ] - | + « N

--------------2 2 7 — cie ( v H----------------5wam2w37>z t= i ^ \ - m1m2map‘s/

Again, from (5.4) and (6.1),

tn < c17m 1m2m 3p i

The number of intervals £„ (concerned with tn’s) therefore does not exceed

ci8 (w1m2m32?2 x*~5T -j- 1).*Hence, from (6.6),

v N — 12 2 Min (v, | cosec tn | )

Mj, m2, = 1 n =■ 0 / 2_6r __^ + a

- c19 2 (m1m 2m 3p 2 x^~:,T -f- 1) ( v + _"m2, m3 = l ' m1m2m3292

c19 5r —{— V4 —f- v3 C(dr + A + a 2 ajVT I + a | 2 —

+ zY-r-t4- x V T-* + a + p -* x V T-* + aa)

C21 ( x ^ T~ i + a + p~2 X? ~ i + 2a).

1 ,3d

This is (6.3), and Lemma 5 is established.7. Returning now to Lemma 4, let 0 2 We take

q = [v V - ? s)] + i (0 

Q = 7> + 1 (72xT 3721 < c22»T.t| aT < qr < 2xt xt/72 < q < 72:cT

Now to every existent q-sum in the left side of (7.1) corresponds a set of numbers p,Q, Q' satisfying the conditions of Lemma 5, which gives therefore

2 2 eiz0 < V ^ j 2 x T Q'

< c x 2 (ztV + T5 + “ _j_ p ~ i x ^ r - + “)0 Q, is, of course, interpreted to be zero.

Tence, from (7.1)

7.2)

We write now

2 r (n) eiz Vn I < c25 + «lo+.<-r < n < 4x2T

af = af° — 2xTl — 4af* = ... = 2mxTm,

where M is determined by2 > xTu > 1.

Taking t = r,n in (7.2), we have for M > 1

r exzJnMs

m - 1 2rnif m < n < 4x:2Tm< c25 2 x ^ T,n + L + a

m = 1

< c25Ma;li»T + 7o + J < c265ci*t + 7o +or

I 2 f (n)eiz 'Ja | < cmx^>T + **+ -f c27 == c2Bx ^ T + L + 2«.I 1 < n - x2T

Writing y for x 2 t in this, we have

J 2 r (n) eiz j < c2Sx*'6 + 2a (1

I t follows by partial summation th a t

2 ei z j n < c29x ^ + 2a 2l<n<x'5V+€

— 30^2 + • e + 2a

If in this we take a = fe we obtain the result of Lemma 4, and our proof is completed.

Note on the Preceding Paper by Prof. E. Landau.— Added August 2, 1924.

8. The authors deduce, by means of the ideas of Lemma 5, their main result from the formula of Lemma 1. I t is possible to s tart from the relation

(8.1) [ {R (y) — 7ty} dy — — c31a# 2 - M cos (2tc i^c) - f (&•'),Jo n=l

which is classical. I am able, indeed, to refine on the authors’ x€, and in fact to deduce from (8.1) th a t

R ( x) — 7t x — 0 lo g ^ x).

488 The Lattice Points of a Circle.

The proof of Lemma 5 provides the inequality

^ ^2tri^y^/ (p2+q2)

q = Q<C32 (y * Q * log4 Q + y ~ * p '4Q ^ logl Q)

for ? /> l , 1 ^ ? > < Q -Q /<C2Q, whence, by Abel’s Lemma, with § or

(8.2)

Q' fin iVyV(p2+q2)2 < c 32 (yQfo-2jlog" Q + y ~ ^ p ~ iqn-2 j logi Q).

? = Q (p2 + f Y

Now (8.2) is valid without the assumption Q' < 2Q, provided we replace

c32 by c33.Therefore, if 2 a< yp

r (n) e 2ni J n y < «3 l + 8K rs V Ib )

e 2ni *jy J (p 2+q2)

( ? '2 + 9 2) 4

(8.3)

V (^ ) ^2triy/ny

n > a>

- c34 + c35 2 (y8°p °log*p + y gl°p *°log®p)1 — P — */(§<■>)

< c 36 («/8‘°co log4 y + y “ * to44 log4 y) < c 37y * to4 log4 y,

g2ni *Jy J (p2 -p q2)c38to 7* -}- 8 2

^ = 1 lg>M ax. {^((o-p2),^} (P 2 + ?2) 4

< e 38to * 4* c 39 (y®°co ®log®o)-|-2/ 80P *<o log4 to)

+ c40 2 (yLp~i log4 -f log4p)P > a/ ( i <*>)

(8.4) < c41 (to 'f -f yvoto'*5 log4 y -f y~*to~t4 log4 y) < c42y*to~* log4 y.

I now taketo = a;44 lo g '* x, log* x.

Observing that i 2 i* ( ( 1 ——

2 ~ — “ {y*cos (2tt s /n y — Jtc)} dy( wS dyCX ± Z

- C« L ,i s — sin (271 \ /n y — |7r) + 0 (za: 4),

we obtain, from (8.1), (8.3) and (8.4),

f {R (y) — Try} dy = ( R (y) dy TJ X Jx

=0 (za;4a:*to4 log4 cc) + 0+ 0 (a^a^to-75 log8 -f- (®4

= 0 (z2),

tz x -\-0 (z)-^^- [R (y) dy £ R {x) - f R (y) dy ^ n x 0 (z).Z J x — z Z J x

thelattice points of a circle.rspa.royalsocietypublishing.org/content/royprsa/106/739/...0 — n —...

Documents