Glasgow Math. J. 42 (2000) 157±166. # Glasgow Mathematical Journal Trust 2000. Printed in the United Kingdom THE MEAN SQUARE OF THE LOGARITHM OF THE ZETA-FUNCTION MICHEL BALAZARD Laboratoire d'Algorithmique ArithmeÂtique (CNRS), MatheÂmatiques, Universite de Bordeaux 1, 351, cours de la LibeÂration, 33405 Talence, France e-mail:[email protected] and ALEKSANDAR IVIC Katedra Matematike RGF-a, Universiteta u Beogradu, Djusina 7, 11000 Beograd, Serbia (Yugoslavia) e-mail:[email protected], [email protected] (Received 15 December, 1997) Abstract. We investigate the function R T; , which denotes the error term in T 2 the asymptotic formula for 0 log it dt. It is shown that R T; is uniformly bounded for 1 and almostj periodi c injthe sense of Bohr for ®xed 1; hence R 1 R T; 1 when T . In case 2 < < 1 is ®xed we can obtain the bound 9 2 =8 " ! 1 R T; " T À . 1991 Mathematics Subject Classi®cation. Primary 11M06. It is well known that the logarithm of the zeta-function may be written as the absolutely convergent Dirichlet series s 1 s 1 1 1 log s log 1 p À log 1 p À ; 1 À À kpks p À p À k 1 p Y X X X where henceforth p denotes a prime number. This expansion is valid in the half- plane Res > 1, and therefore log s is almost periodic in this region. In particular, for 1 and < T < , let À1 1 T 1 1 1 log it 2dt T R T; : 2 k2 p2k 0 j j k 1 p Z X X The function R T; may be thought of as the error term in the asymptotic formula (2), and it is classical that R T; o T as T grows to in®nity, for every ®xed > 1. j j The behaviour of log is much more mysterious when s lies to the left of the vertical line Res 1. For instance, the Riemann hypothesis is obviously equivalent 1 to log being holomorphic in the quarter-plane Res > 2, Ims > 0, and it is an important task to investigate this function there. A natural question is: does log 1 retain some kind of almost-periodicity in the half-plane Res > 2, in spite of possible singularities due to zeros of (and of course the pole s 1)? This is, for instance, dealt with in an important paper by Borchsenius and Jessen [4]. Our contribution to this topic is the study of the mean-square of log on vertical lines in this half-plane. We may consider only the values T > 0, since log s log s . Curiously we were not able to ®nd anything relevant in the literature concern ing the estimation of 1 R T; ; (nevertheless, see A. Selberg [13], where the mean square of arg 2 it is evaluat ed as well as moments of even order, A. Fujii [6], where the mean squ areof 158 M. BALAZARD AND A. IVIC 1 arg i t h arg it is evaluated for 2 < 1 and F.T. Wang [16] for a related problem ). ÀOur most precis e result concerns thebehaviour of R T; in the case in which 1 is ®xed, and is contained in the following theorem. Theorem 1. The function R T; is uniformly bounded for 1 and < T < . Moreover, if 1 is ®xe d, then the function R T; is almost peri- odicÀ1 in the sense1 of H. Bohr. Corollary. If 1 is ®xed, then the function R T; has no limit when T ; hence R T; 1 . ! 1 The corollary is an easy consequence of the theorem and the fact that R T; is not a constant (because log it is not a constant). For the theory of almost periodic functions, we referj the reader j to the book of H. Bohr [3]. One may wonder if R T; is uniformly almost periodic with respect to 1. We cannot answer this questio n. As we shall see, the proof of Theorem 1 follows from an application of Hilbert's inequality (in the version of Montgomery-Vaughan [12]) and the convergence of the series 1 1 ; 3 n 1 pn pn 1 pn X À where pn is the n-th prime. 1 Let us note that the de®nition of R T; makes sense also if 2 < < 1. The double series in (2) is clearly convergent in this range. Furthermore log s is ana- lytic in the complex plane from which the countable union of half-lines sk ; sk À 1 has been removed, where sk runs through the zeros of s , or sk 1. Further, when is ®xed, the function log it has discontinuities of the ®rstkind or, at worst, 1 logarithmic singularities, and so R T; is a well de®ned function of T for > 2 1 ®xed. However, if 2 < < 1 we are unable to obtain results as sharp as when 1, and in this case we shall prove a weaker result, given by the following theorem. 1 Theorem 2. Let 2 < < 1 be a ®xed number. For every positive ", we have 9 2 À " R T; " T 8 : 4 We remark that our theorems could be generalized to the mean square of log F s for suitable Dirichlet series in the ranges a and > , respectively, where a is the abscissa of absolute convergence of F s , and Res is the ``critical line'' for F s . An example of such a class of Dirichle tseries is given by A. Selberg [14]. His Theorem 1 on p. 373, specialised to the case F s s , leads to the asymptotic formula T 1 2 log 2 it dt T log log T O T log log T : 0 j j Z p 1 Finally if one wishes to evaluate the left-hand side of (2) for < 2, then this case can 1 be reduced to the case > 2 by the use of the functional equation THE LOGARITHM OF THE ZETA-FUNCTION 159 s s 1 s s s 1 s ; s 2 À sin À 1 s : À 2 À Proof of Theorem 1. We begin by proving that R T; is uniformly bounded for 1 and < T < . Simple continuity and parity arguments show that it is enough to considerÀ1 the1case > 1 and T 0. By (1) we have 1 1 log it : kpk it k 1 p X X In order to evaluate the integral of log it 2, we use the following lemma, due to Montgomery and Vaughan (see [j11, (28) p. 140]j and [12]). Lemma 1. Let an n1 1 be a sequence of complex numbers and n n1 1 a sequence f g f g 1 of real numbers such that an < and n : inf m n > 0, for every n. Then n 1 j j 1 m n j À j 6 T P 2 1 2 1 1 a int 2 n ane dt T an O j j ; 0 n 1 n 1 j j n 1 n ! Z X X X where the O±constant is absolute. We remark that the theorem of Montgomery and Vaughan is formulated for ®nite sums, but the hypotheses made in Lemma 1 permit this straightforward gen- eralization. Hence 1 1 R T; ; 5 k2p2k p; k k 1 p X X where (p; q denote primes) p; k min log q` log pk : 6 q` pk j À j 6 On the right-hand side of (5) the terms with k 2 will be obviously convergent. When k 1, (6) gives (p pn, the n-th prime) ` pn; 1 min log q log pn q`;` 1 j À j ` q pn pn rn pn min log À n j À j ; q`;` 1 p pn ` where rn q is the prime power closest to pn. Hence we have to estimate 1 1 : 7 n 1 pn rn pn X j À j Observing that pn rn pn rn, we see that the portion of the sum in (7) for which j À j rn is not a prime is clearly convergent, and the remaining portion is convergent by 160 M. BALAZARD AND A. IVIC the convergence of (3). In fact the convergence of (3) follows from the upper bound in the following estimate: x 1 x log log x 2 S x : 2 : 8 pn 1 pn log x pn x log x X À The lower bound in (8) follows easily from the Cauchy-Schwarz inequality; namely 1=2 x 1=2 1 pn 1 pn S x xS x : log x À Á pn x pn x ! X X For the upper bound we need a sieve estimate (see e.g. Halberstam±Richert [6]). We have that 1 1 À x 1 1 : p 2 p x;p h p p h À log x XÀ 0 Yj Thus, using this estimate we have 1 1 S x 1 1 h h h H pn x;pn 1 pn h h>H pn x;pn 1 pn h X X À X X À 1 x 1 h H log x h H p x;p h p X XÀ 0 1 x x 1 2 H log x h H h p h 1 p Á log x X j À 1Q x x x log H x x log log x ' h 2 H log x 2 H log x 2 h H Á log x log x log x X with the choice H log x.