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 in†jthe 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 † ˆ †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 † ˆ p À † ˆ p À † ˆ k 1 p † Y X Xˆ 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 ˆ k 1 p ‡ † † Z Xˆ 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. † ˆ † 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 ˆ 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 Jessenˆ [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 † ˆ 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 ‡are†of 158 M. BALAZARD AND A. IVICÂ

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- À1odic in the sense‡1 of H. Bohr.  †

Corollary. If  1 is ®xed, then the function R T;  has no limit when T ; hence R T;   1 . † ! 1 † ˆ † 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 † Xˆ ‡ 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 Š 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 ®rstˆkind 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 t†series is givenˆ 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 ˆ ‡ 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 : † ˆ † À † † ˆ 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 the‡1case  > 1 and T 0. By (1) we have 

1 1 log   it : kpk  it ‡ † ˆ k 1 p ‡ † Xˆ 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 ˆ f g ˆ 1 of real numbers such that an < and n : inf m n > 0, for every n. Then n 1 j j ‡1 ˆ m n j À j ˆ 6ˆ T P 2 1 2 1 1 a int 2 n ane dt T an O j j ; 0 n 1 ˆ n 1 j j ‡ n 1 n ! Z ˆ ˆ ˆ 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 † Xˆ X † where (p; q denote primes)

 p; k min log q` log pk : 6 † ˆ q` pk j À j † 6ˆ

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) ˆ ˆ `  pn; 1 min log q log pn † ˆ q`;` 1 j À j  ` q pn pn rn pn min log À n† ‡ j À j ; ˆ q`;` 1 p  pn 

` where rn q is the prime power closest to pn. Hence we have to estimate ˆ

1 1 : 7 n 1 pn rn pn † Xˆ 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  † ˆ 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 ˆ 0 Yj  

Thus, using this estimate we have

1 1 S x 1 1 † ˆ h ‡ h h H pn x;pn 1 pn h h>H pn x;pn 1 pn h X  X‡ À ˆ X  X‡ À ˆ 1 x 1 h H log x  h H p x;p h p ‡ X  XÀ ˆ 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 ˆ h H Á log x ‡  log x ‡  log x X † with the choice H log x. The bounds in (8) are given as (38)±(39) on p. 123 of ErdoÈ s±Re nyi [5], butˆ our proof is di€erent from theirs. Nothing better than (8) seems to be known, and so improving (8) seems to be another interesting problem, which is indirectly related to the mean square of log   it . Now we turn to the proof of the second part of ‡the† theorem. We consider separately the cases  > 1 and  1. In the ®rst case, the function R T;  is the primitive of a function almost periodicˆ in the sense of Bohr, and the fact that† it is bounded ensures its Bohr almost-periodicity, according to a well known theorem of Bohr (see [2, p. 123]). In the second case, the function log  1 it 2 is no longer Bohr almost-periodic (in fact, it is not even bounded). Neverj thele ‡ss †jthis function is almost periodic in the sense of Stepano€ (L1) and the theorem of Bohr mentioned above is still valid in this case, as is easily seen by inspection of the proof; (for the de®nition and main properties of almost periodicity in the sense of Stepano€, see [15]; Bochner [1] is also relevant, especially p. 251). To see that log  1 it 2 is almost periodic in the sense of Stepano€ (L1), it suces to prove thatj log  ‡1 †jit is almost periodic in the sense of Stepano€ (L2). This last fact is a consequen ce‡of the† THE LOGARITHM OF THE ZETA-FUNCTION 161 following lemma, essentially due to Wiener and Wintner (cf. [17]) and suggested to us by the reading of a recent paper of J.-P. Kahane [10].

s Lemma 2. Let F s n1 1 annÀ be a Dirichlet series with nonnegative coe- † ˆ ˆ cients (an 0) and abscissa of convergence 1. Let  > 0 be such that F  it has a limit in L2 ;  when  P1. Then F  it has a limit in the Stepano€ ‡L2-me† tric when  À1. † ! ‡ † ! Proof of Lemma 2. Recall that the Stepano€ L2-norm of a function f is given by x 1 2 ‡ 2 f S sup f t dt: jj jj ˆ

1 Suppose ®rst that  2. By the completeness of the Stepano€ norm, it is enough to prove that ˆ 1 x ‡ 2 2 F 1 it F 2 it dt 1 x ‡ † À ‡ † Z À 2

can be made arbitrarily small if 1 < 1 < 2 < 1 , where is small enough, ‡ uniformly in x. Due to the nonnegativity of the an's and an estimate of H.L. Mont- gomery ([11, Theorem 3, p. 131]), this integral is less than

1 2 2 3 F 1 it F 2 it dt; 1 ‡ † À ‡ † ZÀ 2

and the result follows immediately. If  1, we obtain the same result for the metric 6ˆ 2 x  2 1 ‡ 2 f S; sup 2 À f t dt; jj jj ˆ

Proof of Theorem 2. The method that we shall use bears analogies to the one employed in [9] to investigate large values of  s near the line  1. Suppose that 1 † 1 ˆ <  < 1 is ®xed and T T0 > 0. Choose  such that <  < , and let 2  2 T; 2T T T ; ‰ Š ˆ A † [ B † where T consists of t T; 2T such that  w 0 for Rew ; Imw tA log† 2 T. We start from2 the‰ MellinŠ inversion integ ral† 6ˆ(see (A.7) of [8]) j À j  c i x 1 ‡ 1 s eÀ xÀ À s ds c; x > 0 ˆ 2i c i † † Z À 1 162 M. BALAZARD AND A. IVIC and the Dirichlet series representation (1). For T t 2T ; s  it and X TA with suitable A > 0 we have that   ˆ ‡ ˆ

2 i k 1 1 1 p ‡ 1 log  s w XwÀ w dw exp : 9 2i kpks X 2 i ‡ † † ˆ k 1 p À † Z À 1 Xˆ X  

Let t T , choose such that  < < , and replace the line of integration 2 A † i 2 i 2 in (9) by the contour consisting of  2 log T;  2 log T , i 2 i 2 L i 2 ‰ À À À ‡ Š  2 log T; 2 2 log T ; 2 2 log T; 2 i . Then the integrand will be reg- ‰ularÀ onÆ , and theÆonly poleŠ ‰ofÆthe integrandÆ 1†that is passed is w 0, which is a simple poleL with residue log  s . Note that the singularity w 1 s liesˆ to the left of , and thus need not be consider † ed. We have ˆ À L log  s w log T w : 10 ‡ †  2 L† †

This follows by integration from the formula (see (1.52) of [8] or (2.9) of [9])

0 s 1 † O log t ; 11 s s  ˆ ; Im t 1  ‡ † † † j XÀ j À where   0; s  it; 1  2; t t0 > 0: Thus using (10) and Stirling's formula for† ˆthe gammˆ a-func‡ Àtion weobtain

1 w  log  s w X À w dw X À log T: 2i ‡ † †  ZL It then follows from (9) and the residue theorem that

1 it  log  s annÀ O X À log T t T ; 12 † ˆ n 1 ‡ † 2 A †† † Xˆ

1 k pk k where an p exp if n p , and an 0 otherwise. We have ˆ k À À X ˆ ˆ   2T 2 2 log   it dt log   it dt I1 I2; T j ‡ †j ˆ T ‡ T j ‡ †j ˆ ‡ Z ZA † ZB † say. To evaluate I1 we shall apply the mean value theorem for Dirichlet polynomials in the form given by Lemma 1. We obtain

2T 2 1 it 1 2 1 2 annÀ dt T an O n an : T n 1 ˆ n 1 j j ‡ n 1 j j ! Z ˆ ˆ ˆ X X X

We have THE LOGARITHM OF THE ZETA-FUNCTION 163 k 1 1 1 2p a 2 p 2k exp n k2 À X n 1 j j ˆ k 1 p À Xˆ Xˆ X   k 1 1 2p 1 2k 2  2 pÀ exp O X À ˆ k 1= 2k À X ‡ † k 1 p X †   Xˆ X k 1 1 p 1 2k 2  2 pÀ 1 O O X À ˆ k 1= 2k ‡ X ‡ † k 1 p X †    Xˆ X 1 1 2k 1  p O X 2 ; k2 À À ˆ k 1 p ‡ † Xˆ X and k 1 1 1 2p n a 2 pk 1 2 exp O 1 X 2 2: n k2 À † X À n 1 j j ˆ k 1 p X log X À ‡ †  Xˆ Xˆ X   It follows that 1=2 2T 2T 2 1 it 1 it 1 1  annÀ dt T annÀ dt T T 2X À : T n 1  T n 1 !  ‡ Z ˆ Z ˆ X X

Hence, in view of a b 2 a 2 b 2 2Re ab , (12) yields j ‡ j ˆ j j ‡ j j ‡ 2 1 it  I1 annÀ dt O TX À log T ˆ T n 1 ‡ † ZA † Xˆ 13 2T 2 2 1 it 1 it  † annÀ dt annÀ dt O TX À log T ; ˆ T n 1 À T n 1 ‡ † Z ˆ ZB † ˆ X X provided that 1= 2 2 X T À †: 14  † If N ; T denotes the number of complex zeros  i of  s such that  ; † T and  denotes Lebesgue linear measure,ˆthen‡ †  j j  Á† 2 2 3 3 7 3  7  T N ; T log T log T T 2À  log T T 2À log T; 15 B ††  ‡ †  À  † where we used the classical bound of A.E. Ingham for N ; T . See (11.26) of [8]. The crucial fact is that the exponent of T in (15) is less than 1. W† e obtain

2 2 1 it it annÀ dt annÀ dt O 1 T ˆ T ‡ † n 1 n X log2 X ZB † ˆ ZB † X  X 1=2 1=2 4 it 1dt annÀ dt O 1  T 0 T 2 1 ‡ † ZB †  ZB † n X log X  X @ A1=2 2T 2 1=2  it  T cnnÀ À dt 1  B ††0 T 4 1 ‡ Z n X2 log X  X @ A 164 M. BALAZARD AND A. IVIC with 0 cn d n , where d n is the number of divisors of n. Applying again the mean value theo rem† for Dir ichle† t polynomials we see that the last integral above is

2 1 2 T d n n À  ‡ † n X2 log4 X  X 2 4 2 2 3 4 4 7 T X log X À log X T X À log T:  ‡ †  ‡

Thus we obtain

2 1 it 3  1 2 2 7 annÀ dt T 4À2 T 2 X À log T: 16 T n 1  ‡ † † ZB † ˆ X

Consequently from (13) and (16) we obtain, provided that (14) holds,

1 1 2k I1 T pÀ ˆ k2 k 1 p 17 Xˆ X † 1   3  1 2 2 7 O TX 2À TX À log T T 4À2 T 2 X À log T : ‡ ‡ ‡ ‡ †  

It remains to estimate

2 2 2 I2 log   it dt log   it arg   it dt ˆ T j ‡ †j  T j ‡ †j ‡ ‡ † ZB † ZB † À Á 18 2 3  9 † log   it dt T 2À log T;  T j ‡ †j ‡ ZB † since arg   it log t. We use the formula ‡ †  log  s log s  O log t  i ; s ; 1  2 : 19 j †j ˆ ; t 1 j À j ‡ † ˆ ‡ 6ˆ À   † † jXÀ j One obtains (19) by integrating (11) over  it; 2 it and then taking real parts of the resulting expression. Moreover the‰contrib‡ ution‡ Š of  i in (19) such that <  is log t. Then the last integral in (18) is, by ˆthe ‡Cauchy-Schwarz inequality,  3  9 I3 log T T 2À log T  ‡ with

2 I3 : log s  dt ˆ T t 1;  j À j ZB † j À Xj  log2 t dt  T 1 2T 1;  T 1; 1 j À j À  X ‡  ZB †\‰ À ‡ Š 1 2 3  5 N ; 2T 1 log u du T 2À log T:  ‡ † 1 j j  ZÀ THE LOGARITHM OF THE ZETA-FUNCTION 165

Therefore it follows that 3  9 I2 T 2À log T: 20  †

1 Finally from (17) and (20), given that 2 <  < <  < 1 we have R 2T;  R T;  † À † 5  3  9 2  2 2 9 À 2" TX À T 4À2 T 4À2X À log T " T 8 ‡ ;  ‡ ‡    1=2  1  1 if we choose X T (so that (14) holds),  2 4 2", 2 4 2". Replacing 2 ˆ 1 ˆ ‡ À ˆ ‡ ‡ T by T=2; T=2 ; . . ., " by 2 and adding the resulting estimates we obtain the assertion of the theorem. The optimal value of the exponent of T in Theorem 2 is not easy to determine, since it depends on bounds for the zero-density function N ; T (see Chapter 11 of [8] for a discussion concerning bounds for N ; T ). If the Riemann † hypothesis (that †1 all complex zeros of  s have real parts equal to 2) holds, then the above discussion † 1 " may be considerably simpli®ed. Taking ‡ 0 < " < 2 1 we obtain ˆ 2 À † " 1  1 3 2 2 2 R 2T;  R T;  " T TX 2À T 2X 2À X À : † À †  ‡ ‡ † 2= 3 2 Hence choosing X T À † we obtain, on the Riemann hypothesis, ˆ 4 4 " R T;  T 3À2 ‡ : † " À Note that we have shown that R T;  1 as T for  1 ®xed. It † ˆ † !1 1  would be interesting to obtain an ±result for R T;  when 2 <  < 1 is ®xed, which would perhaps give some indication as to what the† true order of magnitude of R T;  might be. †

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