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$: A. IVIC AND H. J. J. TE RIELE fJo - AMS · 2018. 11. 16. · 304 A. IVIC AND H. J. J. TE RIELE and (3) E(T) = a_lT^ap(f^^/A)\ (D>0), { vaogiogiogr)3/4;] where f(x) = £l+(g(x)) (resp$
mathematics of computationvolume 56, number 193january 1991. pages 303-328

ON THE ZEROS OF THE ERROR TERM

FOR THE MEAN SQUARE OF |£(± + it)\

A. IVIC AND H. J. J. TE RIELE

Abstract. Let E(T) denote the error term in the asymptotic formula for

fJo 2

2

dt.

The function E(T) has mean value n. By tn we denote the n\X\ zero of

E(T) - n . Several results concerning tn are obtained, including tn+x - tn <i il

tn . An algorithm is presented to compute the zeros of E(T)-n below a given

bound. For T < 500000, 42010 zeros of E(T)-n were found. Various tables

and figures are given, which present a selection of the computational results.

1. Introduction

Let, as usual, for T > 0

EiT) = [\^l + it)\dt-Tlo%{PJ-(2y-l)T

denote the error term in the asymptotic formula for the mean square of the

Riemann zeta function on the critical line (y is Euler's constant). In view of

F. V. Atkinson's explicit formula for E(T) (see [2] and [11, Chapter 15]) and

its important consequences, this function plays a central role in the theory of

as).It is also of interest to consider E(T) in mean square, and one has

:\

3v/2?rC(3)

This formula is due independently to T. Meurman [16] and Y. Motohashi [17],

who improved the previous error term cf(T ' log" T) of D. R. Heath-Brown

[10]. One consequence of (1) is the omega result E(T) = Q(rl/4) [6], which

was sharpened by Hafner and Ivic [7, 8] to

(2) E(T) = Q+{r(logr)'/4(loglogr)<3+'OB4>/4

(1) f E2(t)dt = CTi/2 +cf(T\o%T) Ic -- „^J-ff, - 10.3047

x exp(-73v/kSglöilöiT)} (B > 0)

Received August 23. 1989; revised December 20, 1989.

1980 Mathematics Subject Classification (1985 Revision). Primary 11M06, 11Y35.Key words and phrases. Riemann zeta function, mean square, zeros, gaps between zeros.

©1991 American Mathematical Society0025-5718/91 $1.00+ $.25 per page

303

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304 A. IVIC AND H. J. J. TE RIELE

and

(3) E(T) = a_lT^ap(f^^/A)\ (D>0),{ vaogiogiogr)3/4;]

where f(x) = £l+(g(x)) (resp. Q. ) means that f(x) > Cg(x) (resp. f(x) <

-Cg(x)) for some C > 0 and some arbitrarily large values of x ; further-

more, f(x) = Q(g(x)) means that \f(x)\ = Q.+ (g(x)). These omega results

are analogous to the sharpest omega results for A(x), the error term in the clas-

sical Dirichlet divisor problem. This suggests the analogy between E(T) and

2nA(j¿) (see [11, Chapter 15]), which was one of the principal motivations for

Atkinson's pioneering work [2]. However, there is an important difference be-

tween E(T) and 2?tA(^). While E(T) is a continuous function of T (with

derivatives of any order), A(^) is certainly not, since J2„<xd(n) (d(n) is the

number of divisors of n) has jumps for integral x which may be as large as

exp( lo'ff* ). (Here and later, C, Cx, ... denote positive, absolute constants).

From (2), (3) and continuity it is immediate that E(T) has an infinity of zeros,

and the purpose of this paper is to study these zeros and related topics, both

from the theoretical and numerical viewpoint.

It seems expedient, especially from the numerical viewpoint, to study the

zeros of E(T) - n rather than those of E(T). This is because E(T) has the

mean value n . More precisely, Hafner and Ivic [7] prove that, for T > 2,

LfTE(t)dt = nT + 2-V2Yi-l)n^U™n*M)2 n<N V" V V 27 y

(4)

1 -1/4

+ - ún(f(T,n))

where

2nn 4

-2Yd-^(^Y\in(Tlog^-T+«-*-^, s/n \ 2nn I \ 2nn 4

+ cf(Tl/4),

f(T, n) = 2rarcsinhW—; + (2nnT + n n ) - n/4,

arcsinh x = log ( x + y/77i).

AT < N < AT for any two fixed constants 0 < A < Ä. Note that formal

differentiation of the sine terms in (4) leads to Atkinson's formula [2] for E(T)

itself (without the error term if (log T)). Also, on simplifying (4) by Taylor's


THE ERROR TERM FOR THE MEAN SQUARE OF \C.(\ + it)\ 305

formula, one may deduce

rT °°

j E(t)dt = nT + 2-l/\-y4Ti/4Yi-i)"din)n~5'*™{^^-7¡)

[5 n=\

+ cf(T2ß\o%T),

which we shall need in §3. A nice feature of (5) is that the series is absolutely

convergent, so that

LT

(E(t)-n)dt = cf(Ty*)2

and, as shown in [7], the above integral is also Q±(T /4).

The plan of the paper is as follows. In §2 we shall study the general problem

of gaps between consecutive zeros of E(t)-f(t), where f(t) (< tx'A~n for any

fixed 0 < n < \) is continuous. In §3 we turn to zeros of E(t)-n and show, by

using (5), that E(t) - n always has a zero in [T, T + cVT] (c > 0, T > T0).

Some other results involving the zeros of E(t) - n , E'(t) and related topics, are

discussed in §4. In §5 we describe the algorithms we used for the computation

of E(t) - n and its zeros, including an estimate of the errors involved. For

t < 500000, 42010 zeros of E(t) - n were found. In §6 we present tables and

figures with a selection of the results of the computations. These results eluci-

date the behavior of E(t) - n, but clearly much more extensive computations

will be needed to examine the most important conjectures concerning the order

of E(t) - n and the distribution of its zeros.

2. Gaps between general zeros

In this section we consider the zeros of the general function

Ef(t):=E(t)-f(t),

where we shall assume that /(/) is continuous for t > t0(f) and satisfies

f(t)=t?(ti/4-i)

for any fixed n such that 0 < n < 1/4 (note that the sign of /(/) is unimpor-

tant). From (2)-(3) one has trivially E(T) = Q.±(T ), hence also Ef(T) =

Q±(7"1/4), so that by continuity each EAt) has infinitely many distinct zeros

in (i0(/), oo), which we shall denote by /,(/) < t2(f) < . Our aim is to

estimate the quantity

(6) K(f) = inf{c > 0: tn+x(f) - tn(f) « fn(f)},

or, in other words, to estimate the gaps between consecutive zeros of Ef(t),

since (6) implies

tn+l(f) - tn(f) « (tn(f)ff)+E (n > n0(e,f))

for any e > 0. Here and subsequently, a(n) < b(n) means that a(n) =

cf(b(n)), n —> oo. Determining the exact value of K(f) for any / seems a



difficult problem. Our main result on n(f) is contained in

Theorem 1. Leta = inf{c>0: £(r)<fc}.

Then1

(7) a < K(f) <

Proof. Note first that from known results on a (see [11, Chapter 15] and [7])

one has1 139!<«< -^ = 0.324009324... ,4 - - 429

so that unconditionally x(f) > \ . Since, in analogy with the classical conjecture

A(x) C x ' +e for the divisor problem, one conjectures that a = |, perhaps

even x(f) = \ for all /. If true, the last conjecture is very strong, since it

implies [11, Chapter 15] that C(j + it) « tlß+E, which is not proved yet.

Now we turn to the proof of the lower bound in (7). Suppose that a > K(f).

Then for s > 0 sufficiently small, Ef(t) must vanish in [T, T + T"~E] for

T > T0(e), which we shall presently show to be impossible. By the definition of

a , there exist arbitrarily large T such that for any given e > 0 we have either

In both cases the analysis is similar, soo-e/2E(T) > T"~£/¿ or E(T) < -T

we shall consider only the former case. From

E(T + H) - E(T) = |^+ t(\ + it) dt-t{.

it follows that for some absolute C > 0,

(8) E(T + H)-E(T)>-CH\ogT

log- + 2y-lT+H

J>2, 0<H< IT).

Let 0 < H < T"~ Then

EAT + H) - EAT) = E(T + H) - E(T) + c?(T1/4—r/.

> -CT" £\ogT CXT1/4-Í,

which implies

Ef(T + H) > r"-e/2 CT" £iogr 2c,r1/4 " >C2T" -e/2

>0

4 •for some C, , C2 > 0, 0 < e < 2(a - \ + r¡) and T > TQ(e), since a >

Therefore, Ef(t) does not vanish in [T, T + T"~e], which is a contradiction

if a > K(f) and e > 0 is sufficiently small.

To prove the upper bound in (7), suppose that

(9) fl2\ogT <H < Tl,2+\

Assuming that Ef(t) does not change sign in [7", T + H], we shall obtain a

contradiction with suitable H, which will yield K(f) < 5 . If Ef(t) does not

change sign in [T, T + H], then

pT+H

(10) / \Ef(t)\dt =

rT+H

J Ef(t)dt


THE ERROR TERM FOR THE MEAN SQUARE OF |f(| + it)\ 30?

From (1), (5) and (9) we infer that

(11) j + E2f(t)dt=(^C + o(l)^Tl/2H (T^oo)

and

(12) j + Ef(t)dt<^TV4,

since by hypothesis f(t) <t[/4~'î and, for T <t <T + H,

E2f(t) = E2(t) + ^(r1/4"V(0l) + @(Txl2~2n).

We shall use Holder's inequality for integrals, letting 0 < 6 < ^ be a fixed

number which may be chosen arbitrarily small. We have

•T+H , rT+H

HT"2 < ( E2(t)dt= f Eâf(t)E2fâ(t)dt

(fi^oi^ffv^,,™,,;<« Ti6l4IX-S

where we used (10) and (12), and where

/ = j™ \Ef(tf-mx-â) dt = f"" \Ef(t)\MI(X-S) dt.

We need an upper bound estimate for / , and to this, we shall use the large values

technique discussed in Chapters 13 and 15 of [11]. Namely, let T]' <c V <

r1/3 and let R0 = R0(V, T, TQ) be the number of points t. in [T, T + T0]

such that \Ef(t/)\ > V and \t¡ - t}\ > CV for i ¿ j and any fixed C > 0.

Then

\E(t,)\ > \Ef(tt)\ - \f(tj)\ >V- CxTl/4-" > V/2

for i = 1, ... , R0. Analogously as in (13.66) of [11], we obtain

(14) Ä0< TE(TV~i+ R0T¡/9Tin8V~2)

for any given e > 0 . This gives

for

(16) r/>Ci7£/36r7/36+£/2 (C,>0).

In our case, TQ = H > Tl/2\og6T, thus V > Tl/4 holds trivially if (16) is

satisfied, and V > T is impossible since a < 5 . In / we divide the interval

of integration [T, T + H] into subintervals of length V (except perhaps the

last such subinterval, which may be shorter, but whose contribution to / is

clearly negligible), and write / = /,+/,. In /, the maximum of \E(t)\ in each

(15) Än«r1+£F 3



of the subintervals is at most CXH T , while in I2 it is larger. We

estimate /, trivially, using (11), and obtain

/ < fTTl/2(H4ß6T7ß(>+£/2)a/(l~a)

To bound I2, we use the large values estimate (15) (since (16) holds), consid-

ering separately subintervals with even and odd indices, so that \t¡ —1¡\ > V

is fulfilled. Hence, by the definition of a, we obtain (considering cf (log T)

possible values of V of the form V = 2m)

/,«logr max RaV3+s/{l-S)TU4<V<T'+C

~l+£. rj. T/<5/(l-<5) „ „,l+3e+«<5/(l-<5)« T logT max V ' « T

l/<7-'+£

if 0 < 8 < \ . Therefore, (13) gives, for 0 < e < \ ,

,,Tl/2 _ „3<5/4r-l-S+E+aô , rrîâ/4 „l-<5 ~(l-<5)/2 „4*5/36 „7á/36+Jc/2n 1 <&. 1 I + I ti 1 tl I

Simplifying, it follows that

(17) H « Tl/2+S{"~[/4)+£ + T-1/2+£ < j-l/2+á(a-l/4)+£

since a > \ . Thus, if we take

„ „l/2+<5(a-l/4)+2£tl = 1

then for ô and e sufficiently small, (9) holds but (17) is impossible. This

contradiction shows that K(f) < \ , as asserted. D

3. Zeros of E(t) - n

The upper bound in Theorem 1 applies to the case when /(/) = n , that is, to

the zeros of E(t) - n . Henceforth, we let tn = tn(n), so that 0 < i, < t2 < ■ ■

denote the distinct zeros of E(t) - n (these bear no relation to the points ¿( in

§2). Theorem 1 gives

(18) '„+.-'„«'y2+£ in>n0(e)),

but we shall use a special method to improve (18) by removing "e". The result

is

Theorem 2. There exists a constant c > 0 (effectively computable) such that

E(t) - n has a zero of odd order in [T, T + c\fT] for T >TQ.

Proof. We shall use the fact that, if h(t) G C[2, T], and Nh(T) is the number

of zeros of h(t) in [2,7/], then Nh(T) > NH(T) - 1, where NH(T) is the

number of zeros in [2, T] of the function

H(t, a) = uh(u)du


THE ERROR TERM FOR THE MEAN SQUARE OF |CC4 + ") 309

for any fixed real a . For, if T' and T" are two zeros of H(t, a) in [2, T],

then

0 = H(T",a)-H(T',a)= [ uh(u)du.Jt'

Hence uah(u), and consequently h(u), must change sign in [T1, T1'] and have

a zero of odd order in [T1, T"]. In proving Theorem 2, we shall make use of

the asymptotic formula (5), and we shall consider the function

oo

g(t) := ]T¿(«)«~5/4sin(v&7- ^) .«=i

Clearly, g(t) g C[0, oo) , and g(t) has an infinity of zeros, since g(t) = fi±(l)

(follows by the method of Hafner and Ivic [7]). Further, for 0 < H < T, we

have¡•T+H i oo rT+H

/ g2(t)dt=^Yd2in)n~5/2 (l-cos(2v/8^i-|)) dt

n=\

,e-5/4rT+H

JT

eiVSM(s/r7i±s/ri) ^Y imn)£^m,n=\ ;m^n

To estimate the integrals on the right-hand side, we use the simplest result on

exponential integrals (see Lemma 2.1 of [11]): Let F(x) be a real differentiable

function such that F'(x) is monotonie with either F'(x) > m, or F'(x) <

-m < 0 for a < x < b. Then

(19)

Using (19), we obtain

[ e'F(x)

.'a

dx < 4m'

rT+H / 1 °° \jT g2(t)dt= i^Yd2in)n~5/2jH+^i^f)

+i.1/2

,«=1 n<m<2n

oo

£-5/4

1

E« /4E

m - n

,£-3/4'm

n=l m>2nm - n

where

= CH + cf(Vf),

C = If d\n)n-il2 = Z^§, 1.561592,

Hence, for T > TQ, suitable C,, C2, C3 > 0 and Ci\fT < H < T, we have

(20) C,//</;

g\t)dt<C7H.



However,

(T i /2\3/4EiiT):=J2 (E(t)-n)dt = -[-) Ti/4(g(T) + y(T)),

where y(t) is continuous, and by (5),

(21) y(t)=cf(C"n\ogt).

Suppose now that Ex(t) does not change sign in [T, T + H], where H = D\ÍT

for some sufficiently large D > 0. Then g(t) + y(t) also does not change sign

in [T, T + H], and we have

• T+H •T+Hrl+H rl+tl

/ (g(t) + y(t))¿dt< max \g(t) + y(t)\- / (g(t)+ y(t))dtJt t€[T,T+H] JT

rT+H

J (g(t) + y(t))dt(22)v ' ¡-T+H

<c4

Estimating the last integral by (19) and using (21), we obtain

[T+H(23) j (g(t) + y(t))2dt<C5Vf,

where C5 > 0 is an absolute constant. On the other hand, using (20) and (21),

we obtain

/ (g(t) + y(t))2dt= [ (g2(t) + 2g(t)y(t) + y2(t))dtJT JT

rT+H

= J g(t)dt(24)

T 1/12//'/2iogriy g\t)dt)

>CxH + o(H) (T-+00).

Comparing (23) and (24), it follows that

CxH + o(H)<C5Vf,

which is impossible for D > C5/Cx . Hence g(t) + y(t), and consequently

Ex(t), must change sign in [T, T + Dy/T]. By the discussion at the beginning

of the proof it follows that E(t) - n must change sign in [T, T + 2DVT], and

Theorem 2 follows with c = 2D. A more careful estimation of the preceding

integrals (working out explicitly all the ¿f-constants) would yield an explicit

value for c. o

The method of proof of Theorem 2 is fairly general and can be used to yield

results on sign changes in short intervals for certain types of arithmetic error

terms. The key ingredient is the existence of a sharp formula for the integral

of the error term in question (the analogue of (5)). In particular, it follows by


THE ERROR TERM FOR THE MEAN SQUARE OF |f($ + it)\ 311

our method that A(x) changes sign in [x, x + Cxyix] for x > xQ. This also

follows from general results of J. Steinig [22], whose method is different from

ours and cannot be used to yield Theorem 2.

Another important problem is the estimation of tn , the «th zero of E(t) — n ,

as a function of n. Alternatively, one may consider the estimation of the

counting function

K(T):=Y 1-'n<T

Since [T, T + cVT] contains a /„ for T > T0, it follows that K(T) » \fT.

Setting T = tn, we have n = K(tn) > s/t~, or

(25) tn « n .

This upper bound seems very crude to us, and we proceed to deduce a lower

bound for tn , which appears to be somewhat closer to the truth. Note that

K(T) « M(T), where M(T) denotes the number of zeros in [0, T] of the

function

E'(t)[l2+U

[¿ -2v = zV)-log (¿)

Here, as usual, we denote by Z(t) the real-valued function

z(,)=x-"2(j+»)c (±+«)

(*»=«f^)-2v",s'"(¥)r('-s»;

so that \Z(t)\ = \C(j + it)\ and the real zeros of Z(t) are precisely the ordinates

of the zeros of Ç(s) on the critical line Res = j . But M(T) = MX(T)+M2(T),

where MX(T) and M2(T) denote the number of zeros of

t ~ \l/2 ™_ /. t N'/2Z(t)-[\og^ + 2y\ , Z(T)+\\og^ + 2y

in [0, T], respectively. Note that MAT) <L.(T), where ¿-(T") is the number

of zeros of

2/v/log(?/2^)

in [0, T]. It was shown by R. J. Anderson [ 1 ] that the number of zeros of Z'(t)

in [0, T] is asymptotic to ^ log T, and by the same method it follows that

Lj(T) = cf(T\og T). Hence K(T) « Tlog T, and taking T = tn , we obtain

(26) iB»/j/log«.

Our numerical results (cf. Table 2 in §6) indicate that both (25) and (26) are

far from the truth. In the range we have investigated numerically, tn behaves

roughly like n log n , but we have no idea how to prove this in general.


312 A. IVICANDH. J. J. TE RIELE

4. Some further results

As before, let tn be the «th distinct zero of E(t) - n. In this section we

present some further results on the Zn's and related subjects.

First observe that

('„+,-<„), ™* \\ + it

= t[\ogTn+2y-l

¿ r<„+l

+ Eitn+i

\\ + it

Eit„)

dt

= M log 2^ + 27-1 >itn+l-(n)[^g^ + 2y

Therefore, it follows that

(27) max Clj + it - 'l0g ïk + 2y

1/2

This inequality shows that the maximum of |C(j + '01 between consecutive

zeros of E(T) - n cannot be too small, even if the gap between such zeros is

small. On the other hand, the maxima of ]£({■ + it)\ can be larger' over long

intervals. Namely, Balasubramanian [3] proved that

(28) max1£[T,T+H]

U2 +3 log H

^exp UVio^ïoïT/

for (logT)£ < H < T. We recall that the best upper bound for £(5 + it) is,

under the truth of the Riemann hypothesis (see E. C. Titchmarsh [24, p. 354,

Theorem 14.14(A)]),

(29) «i+/,)<CTP(á^

so that the gap between (28) (for H = T) and (29) is not so large.

Another useful inequality is

(30) max \E(t)-n\^(tn+x-tn)\ogtn,'e'„.'„xil

the proof of which is analogous to the proof of K(f) > a in Theorem 1, and

which is actually more precise than the inequality k(u) > a (for /(/) = n in

Theorem 1). Namely, let

\E(l)-n\= max \E(t)-n\.'€['„.'„+,]

Take, for example, H = T = tn= 10" ; then (27) yields max > 6.73 and (28) yields max >

13.47 . Practically speaking, the bounds in (27) and (28) are weak: in [14] we have maxZ(i) =

116.88 ; the corresponding right-hand sides (27) and (28) yield 4.39 and 6.77 . respectively! For

recent results concerning large values of Z(t), cf. [18] and [19].


THE ERROR TERM FOR THE MEAN SQUARE OF \Ç(\ + ¡7)| 313

Suppose E(t) - 7i > 0 (if E(t) - n < 0, then we use

(31) E(T-H)-E(T)<CH\ogT (C > 0, 0 < H < \T),

which is proved analogously as (8)), and use (8). Then

E(~t + H) - n > E(l) - n - CH\ogtn >0

for 0 < H < (E(l) - 7i)/(2C\ogtn). Thus E(t) - n has no zeros in [1,1+ H]

with H = (E(l) - 7i)/(2C\ogtn). Consequently,

(E(t)-7z)/(2C\0èt„) = H<tn+l-tn,

and (30) follows.

Using (30), we may investigate sums of powers of consecutive gaps t x —t .

Namely from ( 1 ) we have, as T —► oo,

¡•2T rt

(32) CxTi/2 -/ E2(t)dt~ Y "t,E2(t)dt.JT ,,. , Jl

7-</„<2r,rl/4iog--7-</„+,-i„ "

The contribution of gaps less than T log" T is negligible by (30) and trivial

estimation. From (32) we infer by (30) that

r3/2 « Y ('«+. - ',) ( ™* ,l*(0 - kI2 + 0(33) T<,,<2T,^-l>T«Uo^T Ve['-'«+l] J

«iog2r Y (t.+i-t*)3 + T-

T<t,<2T.lntt-ln>T"i\0è~2 T

which gives (replacing T by T2~J and summing over j > 1)

-3/2,_-2 ^ ^ y^ ,, t ,3

r„<r

In general, for any fixed a > 1 and any given e > 0,

,(3+r»-£)/4

',,<7'

(34) r3/2iog-2r«^(/„+1-/j

(35) ^+"~£)/4«£,,E ('«+.-',/■

(Here, a(«) <£ n b(n) means that the constant implied by the relation a(n) =

0(b(n)) depends on e and a.) This follows along the same lines as (34), using

• /r,+a/4-£«a£| \E(t)\adt (a>0, e>0)

with a = a - 1 . The last bound for a > 2 (without "e") follows easily from

( 1 ) and Holder's inequality, and for 0 < a < 2 it follows from

1/2 / _ s 1/2

r3/2 « j2TEa/2(t)E2-a/2(t)dt<(£ \E(t)\°dt\ (j*\E(,)\4-adt



on bounding the last integral by Theorem 15.7 of [11]. It may be conjectured

that the lower bound in (35) is close to the truth, that is, we expect that

(36) E('«+.-Oa = ^(3+a+0(,))/4 («>l,r->oo),

tn<T

but unfortunately we cannot at present prove this for any specific a > 1 (for

a = 1 it is trivial).

The lower bound estimate (34) may be compared to corresponding results

f°r yn+\ - yn > where 0 < yx < y2 < ■ ■ ■ are the positive zeros of C(j + it) (or

Z(t)). Large values of y , - yn were (unconditionally) investigated by Ivic

and Jutila [13] and Ivic [12], where it was shown that

(3V) Ei^-yf«T^T,

while from/ \ '/3 / N. 2/3

t« ¿2iyn+i-yn)< E^+.-^)3 E1t„<t \y„<T J \?„<T

« (E^i-^)3) iTlogT)2'3

it follows that

Yiyn+i-yn)3»T\og-2T.7„<T

Thus, it follows that the gaps between the ?n's are, on the average, much larger

than the gaps between the yn's.

If one assumes RH, then (37) may be sharpened. From the work of A. Fujii

(see p. 246 of [24]) it follows that, for any fixed integer k > 1 ,

(38) nog1-" T « Y iyn+\ - yn)k « Flog1-* T,

yn<T

and presumably the bounds in (38) may be replaced by an asymptotic equality.

Of course, the lower bound in (38) is unconditional and follows as in the case

k = 3 which we already discussed.

It is known (see H. M. Edwards [4]) that Z(t) cannot attain positive local

minima or negative local maxima if RH is true. In other words, the zeros of

Z(t) and Z'(t) are interlacing, and on RH, R. J. Anderson [1] proved that the

zeros of Z'(t) and Z"(t) are also interlacing. However, the situation with the

zeros of E(t) (or E(t) - n) in this respect is (unconditionally) quite different.

We have

E'(t) = Z2(t) - (log ¿ + 2y) , E"(t) = 2Z(t)Z'(t) - j ,

E[r](t) = 2Y(r~ 2)zU)(t)Z(r-'-])(t) + (-\r\r - 2)\tl~r

7=0


THE ERROR TERM FOR THE MEAN SQUARE OF |CU + ¿01 315

for r> 3. If E'(f) = 0, then

(39) + it log- + 27.

and heuristically, (39) should hold quite often. Since we have by the remark

following Theorem 1 that x(f) > \ , this would mean that E(t) - n must

have many positive local minima and negative local maxima between large gaps

between its zeros, regardless of the truth of RH. This will be confirmed by the

numerical experiments described in §6. If 0 < t, < t2 < • • ■ are the zeros of

E'(t), let

E = inf{c>0:-r„+1 -t„«<}

and

0 = inf{c>O:y„+1-y„«y;}

denote the exponents for the gaps between the corresponding consecutive zeros

of E'(t) and Z(t). Determining the true values of 6 and £ seems almost

hopeless. Under RH, 8 = £ = 0, and unconditionally, A. Ivic in [11] proved

that 6 < 0.15594583... and in [12] indicated how 6 < 0.15594578... maybe attained. These are the sharpest hitherto published results.

5. Computation of the zeros of E(T) - n

In this section we shall describe how we have computed the zeros of E(T)-n .

We write

(40)

where

(41)

E(T)-7t = En(T) = I(T)-T log ¿ + 27¿71

n,

I(T) M + a dt.

Each time a value of En(T) is computed, the corresponding /(r)-value is

saved, since this can be used in the computation of neighboring En(T)-\a\ues,

in view of the relation

(42) I{T + h) = I(T) + J + t(\ + il) dt.

In §5.1 the formulas used to compute the values of \C(¿ + it)\ are given. The

integral in (42) is computed by means of the Simpson quadrature formula with

extrapolation; this is described in §5.2.

We have developed a numerical algorithm to find as many zeros of En(T)

as possible, starting at T = 0, and proceeding with small steps on T. We

cannot be absolutely sure that this algorithm does find all the zeros, but special

provisions have been made in case of doubt. The algorithm is described in §5.3.

The error control of the computations is explained in §5.4.



5.1. Computation of \C(j + it)\. There are two formulas suitable for the com-

putation of |C(j + it)\ '■ the Euler-Maclaurin and the Riemann-Siegel formula.

The Euler-Maclaurin formula enables us to compute Ç(s) to any desired

accuracy, by taking m and n large enough in

n-\ . 1-5 m

(43) C{s) = Yr + ^n~5 + ^---l+YTk,n(s) + Umn(s),7=1 k=\

where2A:-2

TkJs) = $y«-!

7=0

and

\UmJs)\<s + 2m + 1

m+\,niS)Rt{s) + 2m+\

for all m > 0, n > 1, and Re(s) > -(2m + 1). Here, B2 =-- 1/6, 54 =

-1/30, ... are the Bernoulli numbers. In order to obtain Ç({ + it) to within a

specified absolute tolerance, we may take n sa t/(2n). Thus, the computational

work required is roughly proportional to t. The precise choice we made for m

and n is as described in [20] and [21] (see also §5.4). As an example of the use

of (43), take s = \ (cf. (1)); for m = 2, n = 5 we find £(§) = 2.612375056

with an error which is less than 3.1 x 10" , and for m = 2, n = 6 we find the

value 2.612375259 with an error less than 9.4 x 10~ . For the computation

of C in (1) we took C(|) = 2.612375 (and £(3) = 1.202057).The Riemann-Siegel formula is a substantial improvement over the Euler-

Maclaurin formula for not too small /, since its computing time is proportional

to il/2 rather than t. Write the function Z(t) = x~1/2(j + '0C(j + it) as

Z(t) = exp(i0(/)K Q + ü

where

6(t) = ImUogT (^ + jitJj --t\ogn,

and let x = t/(2n), m = [x ¡ and z = 2(r'' - m) — 1. Then the Riemann-

Siegel formula for Z(t) with n + 1 terms in its asymptotic expansion is given

by

(44)

Z(t) = 2Yk cos(d(t) - tlogk)k=\

+ {-lf-\-^Y^jiz)i-l)Jr-j/2 + Rn(r),7=0

when ^(t) = (f(x~(2"+3)/4) for n > 0 and x > 0. Here, the Oy(z) are certain

entire functions which can be expressed in terms of the derivatives of

_ , , _ , , cos(tt(4z2 + 3)/8)<P0 z = O z = -K—±—--V—!-.

U C0S(7TZ)


THE ERROR TERM FOR THE MEAN SQUARE OF \((\ + it)\ 317

O, and 02 are given by

and. , . <P(2)(z) <P(6)(z)<D2 z =-V +--^ .

2 I6n2 2887T4

The coefficients of rapidly converging power series expansions of <P (z) are

given in [9]. Gabcke [5] has obtained error bounds for Rn(x), for t > 200

and 0 < n < 10. For n < 4 these bounds are optimal, and they are given by

\Rn(x)\ < cy-(2"+3,/4 , where c0 = 0.127, c, =0.053, c2 = 0.011, c3 = 0.031,

c4 = 0.017.

5.2. Computation of I(T + h) from I(T). In order to compute I(T + h) from

I(T) for some step h , we use Simpson's rule with extrapolation as follows. Let

rT+h

(45) I(T,h):=J f(t)dt,

where /(/) = \Ç(j + it)\ . We first compute two approximations /, and 72 to

I(T,h) based on applying Simpson's quadrature rule to the interval [T, T+h],

and to the two intervals [T, T + h/2] and [T + h/2, T + h], respectively:

Ix = \{f(T) + 4f(T + h/2) + f(T + h)}

and

I2 = ^{f(T) + 4f(T + h/4) + 2f(T + h/2) + 4f(T + 3h/4) + f(T + h)}.

Using the technique of extrapolation (cf., e.g., [23, §3.3]), these two values can

be combined to yield the better approximation (provided that h is sufficiently

small):

(46) /«»/a + i/a-/,)/»,

where (I2 - 7,)/15 is a good approximation of the error in I2. This error is

used in our computations as a (very pessimistic) estimate of the error in / .

A possible alternative to (46) might be a Gauss-Legendre quadrature rule. For

example, some experiments revealed that a 3-point Gauss-Legendre rule would

yield roughly the same accuracy as the above 5-point Simpson rule (which,

effectively, is a 4-point rule since the end point value f(T + h) on [T, T + h]

is used as starting point value on [T + h , T + 2h]). However, in order to get

an estimate of the error in the 3-point Gauss-Legendre rule, we know of no

better way (cf. [23, p. 127]) than to apply a 4-point Gauss-Legendre rule, and

compare the results; this would require four extra function evaluations, since the

/-values needed in the 3-point rule cannot be used in the 4-point rule. This is

our motivation for choosing (46). Professor W. Gautschi has kindly pointed out

the alternative of using the 7-point Gauss-Kronrod formula for estimating the



error in the 3-point Gauss-Legendre formula. This also requires four additional

points, but is more accurate than the 4-point Gauss-Legendre formula, since

it has maximum degree of exactness. See, e.g., W. Gautschi, Gauss-Kronrod

quadrature—A survey, in "Numerical Methods and Approximation Theory III"

(G. V. Milovanovic, ed.), Faculty of Electronic Engineering, Univ. of Nis, Nis,

1988, pp. 39-66.

5.3. The zero-searching algorithm. Our algorithm proceeds with a step h to

find zeros of the function E (t), i.e., after the search has been completed for

t < T, the interval [T, T + h] is searched (in certain cases combined with a

second search on [T - h, T]). Now and then, small parts of the computations

are repeated with a smaller, and also with a larger step. This is in order to check

whether the step has to be decreased or may be increased, respectively, in view

of the required accuracy.

Let Tj := jh , L := 7(7}) and Ej := En(T¡) ,7 = 0,1,.... Suppose thatthe interval [0, T¡] has already been treated. This implies that 7 and E- are

known, for j = 0, 1, ... , i. We now compute T(Tj+x) from 7(7^) (by means

of (42) and Simpson's rule as described in §5.2) and then E¡+x (with the help

of (40)).If EjEj+x < 0, then by continuity there is at least one zero between T¡ and

Tj x . This zero is found by a rootfinder described at the end of this section.

If EiEi+x > 0 and Ej_]Ei < 0, then we are finished with the interval

[T„TM].If EjEj+x > 0 and E¡_XE¡ > 0, then E¡_x , E¡ and Ei+X have the same sign.

We check whether |£| < \E¡_X\ and |£(.| < |7i/+1|. If so, this means we have a

local extremum; if not, we are finished on [Tj, Tj+l].

In case of a local extremum, we check whether

(47) \E¡\<h(\og^- + 2y

If not, we know that there can be no zero on [T¡_x, T¡+x] because of

(48) ±EK{t) [ \ + it log¿ + 2y)>-(l0g%- + 2y

and the mean value theorem, and we are finished on [T¡, Tj+X].

If we do have a local extremum such that (47) holds, we fit a quadratic

polynomial through the three points (T;_x, E¡_x), (T¡, E/), (Tj+] , E¡+x) and

compute the point Te where this polynomial has its extremum. If h is small

enough, there should be a zero, T = Tg, of E'n(t) very close to T = Tc. This

zero is found with the Newton process. Next, 7(7/) and Eg := En(Te) are

computed, and if EeEj < 0, then there are zeros on [Tix, Te] and [Te, Tj+l],

which are found by the rootfinder described below. This completes the descrip-

tion of our algorithm, apart from the rootfinder.

The rootfinder is designed to find a zero of En(t) on the interval [a, b],

where E (a) and En(b) have different sign. First, the intersection point


THE ERROR TERM FOR THE MEAN SQUARE OF |{( ' + it)\ 319

(c, En(c)) of the line through (a, En(a)) and (b, En(b)) and the horizon-

tal axis is found. Next, a quadratic polynomial is fitted through the three points

(a,En(a)), (c,En(c)), (b, En(b)), and its zero on [a,b] is taken as the start-

ing point to find a zero of En(t) on [a, b] with the Newton process.

5.4. Error control. The numerical computations were carried out on the CDC

Cyber 995 computer of SARA (the Academic Computer Centre Amsterdam),

which has a floating-point mantissa of 48 bits, i.e., a machine accuracy of about

14 decimal digits.

Our aim was to compute as many zeros as possible of the function En(t)

on the interval [0,5x10], each with an absolute error of about 10~ . This

means an accuracy of at least 5 decimal digits for the smallest zero, and 10

decimal digits for the largest zero below 5 x 105.

The error in the computation of \Ç(j + it)\ was controlled as follows.

For t G [0, 5 x 10 ] we applied the Euler-Maclaurin formula (43) in single

precision. If we assume \Um n(\ + it)\ < 10" , then it follows (cf. [20, pp.

151-152]) that/<•> \—' i r\A/(2m+2),. . ,n ss (2%) 10 (t + m+l).

This still leaves freedom to choose one of either n or m , given / and A . We

took A = 15 and (for most i) m = 100, so that n « 0.2244(? + 101). The

actual error is dominated by the machine errors in the computation of the terms

j-( I +") m (43) a pessimistic upper bound for this error is 10~14/« , and for

the value of n given above, and for t < 5000, this is less than 5.8 x 10

For / g [5 x 10 ,5x10] we applied the Riemann-Siegel formula (44) with

n = 3, in double precision (i.e., with an accuracy of about 28 decimal digits),

and the result was truncated to single precision. We denote this numerical

approximation of Z(t) by Zd(t). An extensive error analysis for t G [3.5 x7 8

10 , 3.72 x 10 ] is given in [15]. A similar analysis shows that for t G [5 x

10 , 5 x 105] the error is dominated by the inherent error in (44), i.e.,

\Zd(t)-Z(t)\ <0.03ir225 < 1.5 x 10"10 for?e[5x lo\ 5x 105].

In order to get an idea of the actual error, we computed \t,(\ + it)\ by (43) and

compared it with |Zrf(r)|,for t = 4900(0.1)5100. The maximum difference we

found was 3.9 x 10~10 at t = 5067.2 .

Since the function En(t) measures how well the integral I(t) is approximated

by the function t(log ^ + 2y - 1) + n, we can expect a loss of significant digits

when we subtract the two terms for the computation of E (t). Therefore, we

computed the integral I(t, h) in (45) so that its contribution to the total error

in I(t + h) (= 7(/) + I(t, h)) was as small as the machine accuracy allows.

Thus, the number h was chosen such that

(49) V1'/,'5 »■»-"•v ' I(t + h)


32!) A. IVIC AND H. J. J. TE RIELE

(Recall that (72 - 7,)/15 is a very pessimistic estimate of the quadrature error

in I(t, h).) Actually, we took h = 0.01 for 0 < t < 500, h = 0.02 for500 < t < 2000 and h = 0.05 for 2000 < t < 500000. Several spot checkswere carried out locally for smaller values of h. To summarize, we estimate

that the number of correct digits in our computation of E (t) varies between at

least 13 decimal digits near t = 0 and about 7 near t = 5 x 105. The absolute

error is about I0~nt(\og¿¡ + 2y - 1) « 5.7 x 10"6 for / = 5 x 105.

In the rootfinder used in the zero-searching algorithm described in §5.3, the

Newton process was iterated to machine precision. Usually, no more than two

Newton iterations were needed for this purpose. The influence of the error in

En(t) on its zeros may be quantified as follows. Suppose that in the neigh-

borhood of a zero t = tQ of En(t) we compute with E%(t) rather than with

En(t), where En(t) = En(t) + e, e being a fixed small number. Then the

Newton process for the computation of t = t0 is given by

fM t¡ Ên(ti)_ t¡ EJt*) eE'n(t') E'n(t') E'n(tl) '

so there is a systematic error e/E'n(tl) in the computation of the zero t = t0.

In particular, when E'n(t) is small for / close to t0 , then the error in this zero

may be large. We found

1max —j—— « 3.015,

r<500000, £„(0=0 \E (t)\

where the maximum is assumed for t = 137538.499969. For e = 10~ , this

means a maximum absolute error in the zeros of En(t) of about 3 x 10~ .

6. Results and conjectures

In this section we present a selection of our computational results. We have

found 42010 zeros of the function En(t) on the interval [0, 500000]. The first

100 of them are listed in Table 1.

For selected values of n , Table 2 compares log tn with log n , and tn with

«log«. The quotient log tj log« is slowly changing, with a global tendency

to decrease. We believe it converges to 1, although very large tn will certainly

have to be computed in order to corroborate this. The quotient tn/n log« first

decreases to 0.8904, and then increases slowly to 1.1180 ; no possible conclu-

sion about a limit is apparent from these data. Perhaps «log« is just a rough

approximation to tn , much as « log « is a rough approximation to pn, the «th

prime.

Data on gaps between consecutive zeros of En(t) are shown in Tables 3, 4, 5

and 6. It appears that the gaps dn := tn-tn_x, « = 2,3,... behave in a very

irregular way. Although we cannot exclude the possibility that k = K(n) = j ,

this seems unlikely to us. In fact, we believe k = \ to hold. Maxima and

minima of the quotient dn/tlJ4{ are presented in Tables 4 and 5, respectively.


THE ERROR TERM FOR THE MEAN SQUARE OF |£(i + ¿f)| 321

Table 1

The first 100 zeros of En(t)

tn tn tn

1

2

34

5

67

89

10

11

12

13

14

15

16

17

18

19

20

21

2223

24

25

1

4.

91317

222731

35.40

45

50

51

52

54

56

63

69

73

76

81

8590

95

97

199593

757482

117570545429685444

098708706900884578337567500321610584

514621658642

295421

750880819660010778

178386799939909522

138399065503665198958639

460878

2627

282930

31323334

35

3637

3839

40

41

42

43

44

45

4647

48

49

50

99.048912

99.900646101.331134109.007151116.158343

117.477368119.182848119.584571121.514013

126.086783130.461139136.453527

141.371299

144.418515

149.688528

154.448617159.295786160.333263

160.636660171.712482

179.509721181.205224182.410680182.899197

185.733682

51 190.52 192

53 19954 21155 217

56 22457 22658 22959 23560 23961 245

62 256

63 26264 26765 280

66 28967 290

68 29469 29770 297

71 29872 29973 30874 314

75 316

809257450016

646158

864426647450

290283323460548079172515635323494672571746

343301822499

805140701637.222188

.912620,288651.883251

,880777.919407

652004683833

505614

76 31877 319

78 32179 32680 330

81 33582 33983 34384 349

85 35486 35887 371

88 384

89 39090 39691 39992 402

93 40694 40895 417

96 43097 43498 43999 445

100 448

788055913514

209365203904978187

589281871410370082890794639224371624554495

873869001409.118200

102390212210

737516735190047725

962383927645425963.648250

037348

Table 2

Some data concerning the order of t

n tn log cn/ log n tn/n log n

25

10

2050

100

200

500

10002000

500010000

2000042010

417

40

76185448978

27666174

13807

3931089563

204737499993

757482685444

500321

909522733682037348559572

.863752

.307534257919

200279

343441

805598

656034

2.25021.7849

1.6075

1.44961.33551.32571.29971.2753

1.26351.2542

1.24211.2380

1.2349

1.2326

3.43182.1977

1.7589

1.28370.9496

0.97290.92350.89040.89380.9083

0.9231

0.9724

1.0337

1.1180

c tlJ4, logtn-n-\

,, which in general is

Combined with the data in

For 4 < « < 42010 we observed that dn

close to best possible, in view of (2), (3), and (30)

Table 5, this supports the conjecture that k(tc) = \ (where k is defined in (6)).

The data on gn and gj log tn in Table 3 support (36) for a = 2 . Table 6 gives a



Table 3

Various data related to the gaps between consecutive zeros

^~j^aäzs: tñh"T.^A W°g'nin :=t.-t.- log¿n/log tn logdn/logn gn

2

5

10

20

50

100

200

5001000

2000

500010000

2000042010

3.5578894.140015

5.1627543.109583

2.8344852.389098

0.0759803.6248240.753268

0.596044

7.98303322.172542

1.240345

1.636594

3.24841.1249

0.86850.3620

0.20960.11320.00240.06900.0096

0.0051

0.0403

0.0741

0.00270.0023

3.39962.1580

2.11751.0609

0.77080.5200

0.01360.5000

0.0850

0.05500.56701.2818

0.0583

0.0615

0.8137

0.4945

0.44350.2612

0.19940.1427-0.3743

0.1625-0.0325

-0.0543

0.1964

0.27180.0176

0.0375

1.83100.8827

0.71290.37870.26630.1891-0.4864

0.2072-0.0410-0.0681

0.2439

0.33650.02170.0463

1.80161.8864

1.71651.5146

1.37731.4164

1.38351.2801

1.2895

1.30671.3173

1.39311.4619

1.5505

1.15510.6566

0.46380.34880.26360.2320

0.20090.16150.14770.13710.1245

0.12220.11950.1182

Table 4

Maxima of djtn 1/4

tn ^^T2

370510

11761321

1322

1472

20744224

4692

4848

5006

6058

823017138

18198

21804

23764

39084

41992

2850

74208475

8520

9708

1436532120

3668538070

3951849552

71699

170654

183304

227502

252647457431

757482136994

462567

277407806973

092619104280

.716667209803

.948268

374558

339822122137

441192

832030147130

378144

958173381229

3.55788924.861362

31.291596

42.08575243.841653

44.285645

54.053035

61.75103076.460074

82.898386

88.99070296.093410

104.276659

123.858798

165.382076

169.425143

186.717169

213.951458261.454651

3.39963.73304.2943

4.54104.5751

4.6155

5.4531

5.64655.7148

5.99336.3746

6.81966.9928

7.5724

8.13898.1900

8.5512

9.545110.0549

frequency distribution of the computed values of dn/tj_x , in classes of length

0.1 . For example, we found 10641 values in the interval [0, 0.1), 818 in the

interval [0.9, 1.0) and 1 value (the largest) in the interval [10.0, 10.1) (cf. the

last entry in Table 4). To summarize: 82% of all values are in [0.0, 1.0), 11%

in [1.0, 2.0), 4% in [2.0, 3.0), 2% in [3.0,4.0) and 1% in [4.0,10.1).

Table 7 presents maximal values of \E (t)\ in intervals of length 25000, and

the location of the adjacent zeros. The computed values of En(t)/t

the order results of E(t) as discussed at the beginning of §1.

1/4 confirm


THE ERROR TERM FOR THE MEAN SQUARE OF |Ç($ + it)\ 323

Table 5

Minima of dn/tn1/4

234

5

689

13

143344

159200

301628

10302674

361668418088

118571198727021

U4.7574829.117570

13.545429

17.685444

22.098708

31.88457835.337567

51.65864252.295421

119.584571160.636660

753.427349

978.5595721604.0128273569.014754

6389.01163818818.622459

27076.31467157197.58187070009.242085

110163.040870111649.073447294421.287720

3.5578894.360087

4.427859

4.140015

4.413263

4.1776773.452989

1.144021

0.6367790.4017220.303397

0.2807390.0759800.0636530.062385

0.0380080.037263

0.0311370.0229310.021013

0.0067780.0047890.005105

ääs:3.39962.95222.5481

2.1580

2.1521

1.82091.4531

0.42910.23750.12160.0853

0.05360.01360.0101

0.00810.00430.0032

0.00240.00150.0013

0.00040.00030.0002

Table 6

Frequency distribution of the dn/tj4x-values, in classes of length 0.1

10641 7208 5243 3192 1829 1812 1760 1082 912 818752 591 561 503 415 389 390 373 320 242210 201 196 190

114

6820

2

2

0

0

1

99

4615

2

3

2

0

83

3013

7

2

1

0

81

40

19

41

1

0

195

73

23

13

170

75

3010

6

5

1

1

143 143 130 121

57

3512

52

0

0

45

3613

4

3

0

0

39

19

9

7

1

0

0

54

216

5

0

0

0

Graphs of En(t) and its derivatives are presented in Figures 1-5, which cover

the intervals [0,50], [123400,123600], [456999.4,457431.4], [277514.8,

277661.5], and [495151.35,495321.95], respectively. Figure 2 shows howEn(t) behaves on an arbitrarily chosen interval. Figure 3 shows this function

near the largest observed ¿/„-value (cf. the last entry in Table 4). Figures 4

and 5 show the behavior of En(t) near its smallest and largest observed values,

respectively (cf. Table 7). The function En(t) may increase sharply, but in view

of (8) and (31) we see that it decreases relatively slowly, which is also reflected

in the graphs. The function E'n(t) has sharp peaks which roughly correspond

to large values of \Ç(j + it)\. Note that Figure 5 displays many local extrema

of E (t) in the large intervals between its consecutive zeros.



Maxima of \E (t)

Table 7

in intervals [i x 25000, (/' + 1) x 25000],

and adjacent zeros tn_x, tn

E,(t) eaW71't tn

4

5

6

7

8

910

11

12

1314

15

16

17

18

19

105730.30130061.30152359.00183134.00

221488.30225005.15263358.05

277660.65304718.75328768.95367120.55379395.50415716.60428843.30

457170.60495152.05

-294.972917

342.688448-364.453147-355.682389

367.810105367.801167404.632562

-436.894699-379.461854

489.881453-387.994451

394.115535-381.854476

474.148290

430.409601506.242025

-16.36

18.05-18.45-17.19

16.9516.8917.86-19.03-16.15

20.46-15.76

15.88-15.04

18.53

16.5519.08

11511

13728

15615

18197

213102162824614

2570127807

29574

3249133424

36021369863908441688

105652130060152263182999

221487

225004263357277514304616

328768366950

379394415610428842

457169495151

215612

547162

935681306300

533725360683300497752120030686180260

894252

773446

911107537715.926578

305121

78.840885130.903167

95.831001135.415687

104.953578

104.423051135.522875

146.699293103.513435

134.489310170.362297134.268815

106.341938195.257502

261.454651

170.597059

Figure 1

En(t) and E'n(t) on the interval [0, 50]


THE ERROR TERM FOR THE MEAN SQUARE OF |CU + ¿01 325

- T-123400.0

Figure 2

En(t) and E'n(t) on the interval [123400, 123600]

E^m

I \ i \.\\\ i. f i.|.I.J ,J - 1 ..i ij>J <^44 387 430

E^T)

- T-456999.4

Figure 3

En(t) and E'n(t) on the interval [456999.4,457431.4]



T-2ÍV51 4.8

Figure 4

En(t) and E'n(t) on the interval [277514.8, 277661.5]

- l~j»l^vjuJL-]ILuJi^

E, (T)

136 153 170

T-495151.35

Figure 5

EM) and E'M) on the interval [495151.35,495321.95]


THE ERROR TERM FOR THE MEAN SQUARE OF |C( i + '01 327

Acknowledgment

We wish to thank D. R. Heath-Brown and the referee for valuable remarks

and J. van de Lune for his comments on a preliminary version of this paper.

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Katedra Matematike, RGF-a Universiteta u Beogradu, Djusina 7, 11000 Beograd,

jugoslavija

Centre for Mathematics and Computer Science, P. O. Box 4079, 1009 AB Amsterdam,

The Netherlands

E-mail address: [email protected]


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