sharper bounds - american mathematical society€¦ · 244 j. barkley rosser and lowell schoenfeld...

MATHEMATICS OF COMPUTATION, VOLUME 29, Number 129JANUARY 1975, PAGES 243-269

Sharper Boundsfor the Chebyshev Functions 6(x) and \f/(x)

By J. Barkley Rosser and Lowell Schoenfeld *

Abstract. The authors demonstrate a wider zero-free region for the Riemann zeta func-

tion than has been given before. They give improved methods for using this and a re-

cent determination that the first 3,502,500 zeros lie on the critical line to develop

better bounds for functions of primes.

0. Introduction. As this paper is dedicated to D. H. Lehmer on the occasion ofhis 70th birthday, it is particularly appropriate to remark on Lehmer's long time interestin the application of numerical computation to problems in number theory. In particu-lar, his papers [1956A, 1956B] reporting that the first 25,000 zeros of the Riemann func-tion f(s) lie on the critical line led the way in the application of modern computing ma-chinery to the study of the zeros of this function. In default of a proof of the Riemann hy-pothesis, the best estimates for i//(x) and dix), and hence of tt(x), pn and otherfunctions of the primes, depend on the current state of knowledge of the zeros of f(s).

The present paper is devoted to obtaining improved estimates for 4>{x), the log-arithm of the least common multiple of all integers not exceeding x, and 0(x), thelogarithm of the product of all primes not exceeding x. We reserve for another paperthe application of these results to tt(x) and pn, simply remarking here that they per-mit the deduction of such inequalities as 7r(2x) < 27r(x) for all x > 11 and d{pn) >«log« for all « > 13. We are also able to show (px + p2 + ••• + pn)/n < xhpnfor n > 9, as conjectured by Robert Mandl.

To a considerable extent, this paper represents an up-to-date version of part ofRosser and Schoenfeld [1962], which will hereafter be cited as R-S. We also makeconsiderable use of Rosser [1941]. We assume familiarity with these papers, and shalluse notation and results from them freely.

Rosser, Yohe and Schoenfeld [1969]** announced that the first 3,500,000 zerosof f(s) lie on the critical line. By applying the stronger result contained in Theorem4 of Lehman [1970], we are now able to show that the first 3,502,500 zeros are onthe line. Our computations extended out to Gram No. 3,502,504; between here andthe smaller Gram No. 3,502,483, f(s) behaves very regularly so that all of the Gram

Received July 18, 1974.AMS (MOS) subject classifications (1970). Primary 10-04, 10H05, 10H25, 33-04, 33A40.*Sponsored in part by the United States Army under Contract No. DA-31-124-ARO-D-462,

in part by the Wisconsin Alumni Research Foundation, and in part by the Rockefeller University.Also, acknowledgement is given to the University of Michigan, which made its facilities available tothe second named author during the preparation of this report.

**We take the opportunity to correct a misprint on line 5, p. 71 of this paper. The coeffi-cient of t-7/4 given there should be 2.28 instead of 2.88.

Copyright © 1975. American Mathematical Society

243

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244 J. BARKLEY ROSSER AND LOWELL SCHOENFELD

numbers in this interval separate the zeros of f(s). Letting N{T) be the number ofzeros p = ß + iy of f(s) such that 0 < y < T, there is an approximation F{T) toN{T) given in Rosser [1941, p. 223]. We define A as the unique solution of F{A) =3,502,500; the calculations of Rosser, Yohe and Schoenfeld establish that N{A) =3,502,500 so that N{A) = F{A). We note

(0.1) log.4 = 14.45443 30529 858 •••, A = 18 94438.51224

Throughout this paper, the inequality sign is used in the strict mathematicalsense; if A < B is written, where A and B contain approximations, then with theindicated approximations the inequality does indeed hold. We have not always giventhe very best bounding decimal approximation but have frequently given less strict (butcorrect) bounds which are easier to verify. We occasionally use the sign = to indicateapproximate equality with the approximations being accurate to 1 or 2 units in theleast significant digit shown. Finally, we use z to denote the complex conjugate ofz, Rz to denote the real part of z, and lz to denote the imaginary part of z.

It is our pleasure to express our thanks to John W. Wrench, Jr. of the U. S. NavyCarderock Laboratory for the computations he performed for us. We also wish tothank Dianne Hollenbeck and Emerson Mitchell of the Mathematics Research Centerfor their help with calculations.

1. A Zero-Free Region for f(s). In this section we give such a region whoseform is essentially that of the classical one of de la Vallée-Poussin. The result, statedin Theorem 1, is substantially better than the corresponding result, Theorem 26 ofR—S. The improvement is due primarily to the work of Stechkin [1970B] with otherimprovements resulting from a better nonnegative cosine polynomial and from theknowledge that all zeros p = ß + iy of f(s) for which 0 1. Let

qo(ffo ~ 0 . „ .s . (P 1 \P =-r"> \ = (2a-l)min ,(2a0- l)a \a 2a0-l)

and 0 < R6 < 1. Then

(i.i) R(^+___L_UXR/_L_+__L\.Vs ~b s-l+b/ \so~b s0-l+bJ

If, in addition, lb = t, then

(1.2) min¿R—^,R-1—= ) > P R ( —~h +-!-=V

Proof. Let a = Rb. For (1.1), it suffices to deal with 0 < a < Vz, since


CHEBYSHEV FUNCTIONS 6(x) AND ii(x) 245

otherwise we consider b' = I — b . Let Q be the quotient of the left side of (1.1)by the real part on the right side of (1.1). Replacing r in the proof of Lemma 2 ofStechkin [1970B] by o0 > a, we find that

2a - 1 ^ 2a- 1 = 2a - 1" Fía, t2) " max{F(a, 0), F(a, °°)} max{0(a), 2a0 - 1} '

where F(a, u) and 0(a) are as defined by Stechkin. Inasmuch as

we see that ß>X so that (1.1) holds.To obtain (1.2), it suffices to prove it with the left side replaced by Rl/{s - b) =

l/(a - a). The quotient of this by the real part on the right side of (1.2) is l/0j(a)where

0,(a) =- +-r—— < 0,(0) = -.1V a0 - a a0 - 1 + a ' P

This completes the proof of (1.2), which is essentially Remark 2 of Lemma 2 ofStechkin [1970B].

We define for a > 1

(1.3) o0 = tt{y/8o2 - Ao + I + 1),

,. .s 2a-I (I_J_)i/2U J K° 2o0 - 1 \ 2 4a - 1 + l/(4a - 1)/ 'Then

(1.5) X = p = k0 > l/y/s, o0>a.

For 0 < x < 0.03 we easily verify that

y/~5 < V5 + 12x + 8x2 < V^ + 2.68847x;

putting x = a - 1 and assuming 1 < a < 1.03, we get

(1.6) t < a0 = ^V5 + 12x + 8x2 + \h < t + 1.34424(a - 1) < 1.659,

where r.= Yi{y/5 + 1) is the golden ratio. Also

(1.7) k0< 0.458 if a < 1.03.

If 0 < x < 1, then

(5 + 12x + 8x2)(l+|x)2 (l-§x2)2

< 5 + 20x + 20x2 - 21x3 - 52x4 - 20xs + 23x6 + 23x7 + 6x8

< 5 + 20x + 20x2 = 5(1 + 2x)2.



If 1 < a < 1.03 and x = a - 1, then

1±2*- >^(l+ix\(l_262V5 + 12x + 8x2 V5 V 5 / V 25'

(1.8)>~r=r U +f(>- O" 1.06496(a- 1)4.

The following is the principal result needed for the proof of Theorem 1.Lemma 2. Let P{d) = Z"k=0ak cos kO be such that all ak>0 and P{d) > 0

for all real d. If ß + iy is a nontrivial zero of f(s) such that ß + \h. and 1 < a <1.03, then

■¿ZJ--¿ZT\


Moreover, for 1 < a < 1.03, a typical term C7(p) of the first sum in (1.14) is

o-ß_l-ß _ {o- l){y2-{o-ß){l-ß)}W (a - ß)2 +y2 (1 - ß)2 + y2 {(a - ß)2 + y2} {(1 - ß)2 + y2}

> a-l T2 -(1.03-/3)(1 -/?)^y2+{l-ß)2' 72 + (1.03 - (3)2

If ß = Vi, then since \y\ > 14.13 4725 we easily get

G(p)>0.997 27lA^l-(i1_ß)2 = 0.997 2714(a- 1)rQ +y^).

If j3 # të, then I7I > A; as 72R{l/p + 1/(1 - p )} < 1 always holds, we have

rin^ta uv(l 4- l \ T2(T2 - 1-03)G(p)>(a-l)R(-+rr^j.(72 + i)(T2 + io32)

>0.9973(a-l)R(i + r^:).

Hence the first sum in (1.14) exceeds

0.997 2714(o- 1) • 2 £ R - > 0.0460 6537(a - 1)p p

by (1.15). Consequently, (1.11), (1.14), (1.15) and (1.12) yield

(U6) "7^l, we have by (1.5) and (1.6)



«/Jía_LkR-L.U-liL, 0 ¥= t,

1_+Y(°°_¿ZL_

CHEBYSHEV FUNCTIONS 0(x) AND \p(x) 249

R{Ko^(°o +iy0)--ç{a + Í'T0)|

(1.23)1 - Kn 1 - K0 17 1< -T-9 log lT0l - —T"2 loë(27r) +'"

2 -"0' 2 — 372 ""0o 'We now obtain (1.9) with ß0, y0 in place of j3, 7 by using (1.10) with t = y0,(1.17), (1.22) with t = 70 and 2


For the choicesa = 0.9126 0875, b = 0.2766 1490,

we obtained a value RQ = 17.449 61294 38363 ••• . An extensive computer search byEmerson Mitchell of the Mathematics Research Center failed to find a fifth-or sixth-de-gree cosine polynomial giving a smaller value for R0. Further, the work of French[1966] shows that, regardless of the degree of F(0), R0 > 16.2568; for other resultsconcerning F(0) see also Stechkin [1970A], who essentially shows, for example, thatfor fourth-degree polynomials R0 > 17.174 8395.

To simplify the calculations, we choose a = 0.9126 and b = 0.2766, whichyield

a0 = 11.185 93553 12082 048, a. = 19.073 34400 4352,a2 = 11.676 18784, a3 = 4.7568, a4 = I,

A0 = 36.506 33184 4352, R0 = 17.449 61294 58 ••• ,4 4Z aklog{2n/k) > 52.38865, Z aJk2 < 22-584.

fc=i fc=i

The preceding value of R0 is smaller by about 0.00014 than a value given by Stech-kin [1970B]. Prior to this work of Stechkin, zero-free regions of the present kind hadF-j replaced by the larger R0. The result below improves Stechkin's Theorem 2 notonly by having a smaller value for R but also by the presence of the denominator 17.

Theorem 1. There are no zeros of f(s) in the region

(1.27) o>l-l/(Rlog|r/17|), \t\>2l,

where R =9.6459 08801.Proof. First, suppose ß + iy is a nontrivial zero of f(s) such that ß + Vi;

then \y\> A. We assume that 1 < a < 1.03 so that by (1.7) the coefficient of2^=1 ak \og{2v/k) in the expression for B is negative; hence this sum can be replacedby its lower bound 52.38865. We also replace f'(a0)/f(a0) by its upper bound givenin (1.25). As the resulting total coefficient of k0 in (1.9) is negative, we may replacek0 by the right side of (1.8). With v defined by (1.26), we set

x = a - 1 = i>/logl7/17 Iand observe that 0.4; hence l


From this we easily get

(1.28) ß21, then (1.28) clearly holds. Finally, as RX


(2 {x - l)Ô,(z, x) + (1 + 2/2 - 2/z(l + x)2)/:^, x)

< A:2(z, x) < (x - 1)0!(z, x) + (1 + 2/z)K.{z, x).

Proof. By (2.1)

K2(z, x) - *,(z, x) = |J~ (i - i)^(i) dr.Integrating by parts gives

K2(z, x) - K.{z, x) = (x - l)ßi(z, x) + ±£° A -—^W(r) A.

Then our lemma follows.Corollary. If 0

ÇHEBYSHEV FUNCTIONS d(x) AND t|/(x) 253

We still have the matter of dealing with K.{z, x) and K2{z, x) when x isnear unity. We first consider the case where z is small or of moderate size. For this,with the computing facilities now widely available, numerical quadrature seems the bestprocedure. In the present situation, it suffices to use the trapezoid rule and the mid-point formula, which are (A2) and (A3), respectively, on p. 446 of Rosser [1967].

We observe that by (2.1)

Kv{z, x) = Kv{z, y) + \\yx f- '//z(f) dt.

We take y large enough so that Kv{z, y) can be estimated with adequate accuracyby Lemmas 4 and 5; note that a high order of accuracy for Kv{z, y) itself is notneeded, since it is usually much smaller than the other term on the right. So we desireto estimate

(2.11) \\Sxmdtby numerical quadrature. When v=l, we have

(2.12) f{t)=Hzit),

(2.13) f'it) = -zf2{t2-l)Hz{tY

(2.14) f'{t) = ^ {z{t2 - I)2 - At}H\t).

By Descartes' rule of signs, the polynomial on the right of (2.14) has at mosttwo positive roots. As the polynomial is positive at t = 0, negative at r = 1, andpositive for large t, it must have exactly two positive roots, t. and t2, with t. <1 < f2. For z not too small, the values of f, and t2 are approximately

If we take 1 + q to be t, or t2, then the recursion

(2.16) qn+, = ± 2y/l + qnIWz{2 + qn)}

will converge fairly rapidly to q.To get an upper bound for (2.11), we use the midpoint formula for tt < t < t2,

and the trapezoid rule for the rest of the range of integration. For a lower bound, weinterchange the midpoint formula and trapezoid rule. If the bounds are not as closetogether as desired, use shorter intervals in the quadrature formula.

When v = 2, we have

(2.17) f{t) = tH*it),±It(2-18) f{t) = - ¿{z(r2 - 1) - 2iWz(f),

(2.19) fit) = ¿j {z(f2 - l)2 - At3}Hz{t).



Clearly we proceed in exact analogy with the case v = 1. Note that the twopositive zeros of f\t) are the reciprocals of the t. and t2 obtained above.

We now consider the case for large z. Define

(2.20) w = i-Jt- 1/VÖ/V2.Then

(2.24) K.iz,x)=ÇS~e—2£dw,

(2.21) t = 1 +w2 +wy/2 +w2,

(2.22) dt/dw = 2w + 2(1 + w2)/y/2 + w2,

(2.23) tdt/dw = Aw{l + w2) + 2(1 + 4w2 + 2w*)/y/2 + w2

= Aw2y¡2 + w2 + Aw{l + w2) + 2\y¡2 + w2.

We have of course

,2 dte "*'y

(2.25) K2{z,x)=ÇS~e— 2 t£dw,

where

(2.26) y = (yß-UyjX~)ly/2.By squaring both sides, we verify that for w2 > 0

(2-27) L^l* ♦ * ♦ l] -"

The integrals appearing in the formulas above are the complementary error func-tion. Means for calculating or bounding it are well known; see particularly 7.1.5, 7.1.6,7.1.13, and 7.1.23 of NBS #55.


CHEBYSHEV FUNCTIONS 8(x) AND 0

(2.32) ^=>(,--)/V2,

whence1 +w2

V2"+r)M-

With this and (2.22), we can get a lower bound for Kx{z, x). Integration by partsgives

If» 1 + w2 _zw2 ,e dw.(2.34) rw2JÏ+^2e-™2dw=y-^^e^y2+\ïJy 2z ziy 0 + vv2

With this, (2.23), (2.32), and (2.33), we can get a lower bound for K2{z, x).If we let x go to 0 in (2.30) and (2.31), we get the known results

(2.35) K,iz)


N(y) by F{y) + Riy). Integration by parts gives

Z $(7)

CHEBYSHEV FUNCTIONS 0(x) AND \¡j(x) 257

i I 0(X) - {X - log (27T) - \h l0g(l - X"2)} I(3.11)

< {Sxirn, Ô) + S2im, 5)}¡y/x + 53(m, 5) + S4(m, 5) + mô/2.

Proof. Split the sum on the right of Theorem 13 of Rosser [1941] into foursums over the regions: I, ß < Vi, 0


where z = 2Xy/m, U' = (2m/z)log(t//17), V = {2m/z) log(F/17); we get (3.16) byputting y = 17 exp(zí/2m). Also we get

\Vvy~X^V {-X2/log(y/17)}log £ dy

(3.17)= Xa{T{- 2, v") - r(- 2, {/")} + x2{iogni2n) {r(- 1, v") - T{- i, u")}

with U" = X2llog{U/l7), V" = X2/log{V/n) by putting y = 17exp(X2/í).Theorem 2. // log x > 105, then

(3.18) I0(x)-x|, |0(x)-x|

CHEBYSHEV FUNCTIONS 6(x) AND i//(x) 259

F0 = R{A)4>,{A) = %jP {exp{- X2llog{A/l7)}X*e2X}X-\-2X.

As the expression in the large braces takes its maximum at

_ 1 . A , if. 2 A ,, A\*^ = 2 log T7 + 2 Il0g 17 + l0g 17 j '

we conclude

(3.27) F0 < lO-6X-1/2e~2X.

If Wl>A, then Y = W. and Ar>16. As R(y)/logy is decreasing fory>ee, (3.25) gives

E0 < 2R{Y)l(V2 J■logi4 IV2

so that we conclude (3.27) for this case also. Then by (3.26)

(3.28) Z 0i(7) < 0.01659 38121 {l + ^f8^ +^^-)x^2e-2X.A 105, we see that bX2 is a decreasing function of X; hence 8X2 <0.3277 and 0 < 5 < 1 - 1/x. Hence

(3.31) 2+ 25 +02


(3.32) |0(x)-x|,|0(x)-x| 59, since otherwise e*(x) > e(x) and we can appeal toTheorem 2. In (3.6) through (3.11), we take m = 1, T, = 0, and

(3.36) T2 = l7X~%ex.

As X > 59, we have A< T2 < W0 and If, < F2.We can treat {5,(1, 5) + S2(l, b)}/y/x and the error terms Ej{U, V) arising

from the use of Lemma 7, Corollary, as we did in the proof of Theorem 2. Thus, wecan proceed as though

(3.37) s3{i, ô) 0, then

Tiv, x)

CHEBYSHEV FUNCTIONS d(x) AND \p(x) 261

By (3.16)

(3.43) J~ çt.iy) log ¿ dy = -| {X2/:2(2X, [/') + ^,(2*, £/')},

where

Write temporarily

(3.45) q = 3-^, y = {y/ÏÏ- Uy/Û')/y/2.

Then y is negative, and y2 = AifJ' + \¡U') - 1 =q2/{2{l -q)}. So, by splittingthe integral at w = 0, we get

(3.46) V2Te-2^2¿w/y Vl - q

Hence, by (2.30) we get

(3.47) XdK.{2X, l/)


determined so that if x > eb, then

(4.1) |0(x)-x|

CHEBYSHEV FUNCTIONS 8(x) AND \jj(x) 263

S,{m, b)


S3{m, 5) + S4{m, 5) < Í225_m.

So we use Lemmas 8 and 9.Theorem 5. Let Ty>D and A 0

(4.11) H>, x) = xv~le~x +{v- 1)H>- l,x),

whence for v < 1 we get

(4.12) Tiv, x)

CHEBYSHEV FUNCTIONS d(x) AND \¡>(x) 265

Theorem 6. We have

(5.1) 0(x)< 1.001 102x if 0 0.998x if x > 487,381; 0(x) > 0.995x if

x > 89,387; ö(x) > 0.990x if x > 32,057; ö(x) > 0.985x if x > 11,927.Proof. These follow from (5.2) above, from (4.6) of R—S and from the Appel-

Rosser tables [1961].This corollary supplements Theorem 10 of R—S, and the lower bounds given for

x cannot be replaced by smaller ones.Theorem 7. If x > 108, then

(5.5) |0(x) - x |,| 0(x) - x | < 0.024 2269x/log x.

Proof. If 108


As the material in the braces increases for the x specified, it does not exceed its valueat x = e18'43 which value is less than 0.024 2269. We continue to use the table inthis way until we have dealt with the range e29 - {e(x)log x}x/log x > - 0.021x/log x.

This completes the proof of the lower bound for 0(x) and hence for 0(x). Theproof for the upper bound for 0(x) is easier since the extra terms y/x and 3X1'3do not appear.

Corollary 1. If x > 525,752, then

dix) -x< 0(x) -x < 0.024 2334x/log x.

Proof. We apply (4.12) and (4.5) of R-S.This result may hold for smaller values of x as well. However, in the next corol-

lary, the bounds for x cannot be lowered.Corollary 2. We have

(5.6) |0(x) - x|< 0.024 2334x/log x if 758,699 < x,

(5.7) 10(x) - x |< x/(40 log x) if 678,407 < x.

Proof. The previous corollary takes care of the upper bounds for 0(x). Thelower bounds are handled by using (4.6) of R-S as well as the Appel-Rosser tables

[1961].Theorem 8. If x> 1, then

(5.8) |0(x)-x|, \sb{x)-x\ 108. For 1 < x< 108 we use (4.12) and (4.5) of R-S to get

0(x)-x/3

CHEBYSHEV FUNCTIONS 8(x) AND t//(x) 267

m2

22

2

22

2

2

3

3

34568

911

1112

12

12

121212

12

11

1110

1010

2.69(-4)2.68 (-4)2.67(-4)2.66(-4)2.61(-4)2.45(-4)2.24(-4)1.97(-4)8.47(-5)

5.88(-5)4.61(-5)2.11(-5)1.18(-5)7.75(-6)4.69(-6)3.90(-6)3.05(-6)3.02(-6)2.76(-6)

2.73(-6)2.72(-6)2.70(-6)2.66(-6)2.61(-6)2.53(-6)2.64(-6)2.54(-6)2.67(-6)2.56(-6)2.46(-6)

1.2015(-3)1.1969(-3)1.1924(-3)1.1878(-3)1.1653(-3)1.0800(-3)9.6459(-4)8.0243(-4)6.594K-4)4.4170C-4)3.0007(-4)2.021K-4)1.3730(-4)9.408K-5)6.5642(-5)4.7407(-5)3.5960(-5)2.8876(-5)2.4539(-5)1.8315(-5)1.7748(-5)1.7583C-5)1.7285(-5)1.6993(-5)1.6424(-5)1.5830(-5)1.5257(-5)1.4682(-5)1.4104(-5)1.3548(-5)

4 50

500600

700800900

100011001150120013001350140015001600170017501800185019002000210022002 300240025002700300035004000

m

9987

765

54433

3

32

2

2

22

2

2

22

2

2

22

1

11

2.59(-6)2.48(-6)2.5K-6)2.55(-6)2.28(-6)2.30(-6)2.35(-6)2.03(-6)2.24(-6)2.06(-6)2.16(-6)1.94(-6)1.74(-6)1.4K-6)1.48(-6)1.13C-6)9.82C-7)8.56(-7)7.47(-7)6.52(-7)4.97(-7)3.80(-7)2.90C-7)2.22(-7)

1.70(-7)

1.3K-7)7.75C-8)4.77(-8)1.22(-8)3.42(-9)

With b = 18.42068, m = 2, and 5 = 2.6855(-4), one gets e< 1.20116(- 3).Most entries were calculated by (4.8). The last three were calculated by (4.10).

0(x) - x > - 2.06Vx = - 2.06 ̂ =^ • -^-r-y/x logfcX

> - 2.06 {2k)K >- vk-ek log^X logfcX

for k - 2, 3, 4.Theorem 9. If e(x) is defined by (3.19) or (3.20), then

(5.10) 0(x) - x < 0(x) - x < xe(x) for 0 < x,

(5.11) 0(x)-x> 0(x) - x > - xe(x) for 39.4


x < e105. As e(x) decreases for x > 1, we have e(x) > 0.03 if x < e105. Fromthe table and (5.2), we deduce (5.10) and (5.11) for 108 0.23 so that Theorem 10 of R-S yields 0(x) > x - xe(x)for 101 < x < 149. We readily complete the proof of (5.11) for 39.4 < x < 101.

Mathematics Research CenterUniversity of WisconsinMadison, Wisconsin 53706

Department of MathematicsState University of New York at BuffaloAmherst, New York 14226

MILTON ABRAMOWITZ & IRENE A. STEGUN (Editors), Handbook of Mathematical Func-tions, With Formulas, Graphs, and Mathematical Tables, Nat. Bur. Standards Appl. Math. Series, 55,Superintendent of Documents, U. S. Government Printing Office, Washington, D. C, 1964; reprinted,Dover, New York, 1964. [Cited as NBS #55.] MR 29 #4914; 34 #8606.

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CHEBYSHEV FUNCTIONS d(x) AND i|/(x) 269

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