For a fixed integer $k\ge 3$ and fixed $1/2  1$ we consider $$
\int_1^T |\zeta(\sigma + it)|^{2k}dt = \sum_{n=1}^\infty d_k^2(n)n^{-2\sigma}T
+ R(k,\sigma;T), $$ where $R(k,\sigma;T) = o(T) (T\to\infty)$ is the error term
in the above asymptotic formula. Hitherto the sharpest bounds for
$R(k,\sigma;T)$ are given for certain ranges of $\sigma$. We also obtain new
mean value results for the zeta-functions of holomorphic cusp forms and the
Rankin-Selberg series.Comment: To the memory of R.A. Rankin, 15 page

Ivić, Aleksandar

English

arXiv

Aleksandar Ivić

Journal de Théorie des Nombres de Bordeaux

JOURNAL DE THÉORIE DES NOMBRES DE BORDEAUXALEKSANDAR IVIĆOn mean values of some zeta-functions in the critical stripJournal de Théorie des Nombres de Bordeaux, tome 15, no 1 (2003),p. 163-178<http://www.numdam.org/item?id=JTNB_2003__15_1_163_0>© Université Bordeaux 1, 2003, tous droits réservés.L’accès aux archives de la revue « Journal de Théorie des Nombresde Bordeaux » (http://jtnb.cedram.org/) implique l’accord avec les condi-tions générales d’utilisation (http://www.numdam.org/conditions). Toute uti-lisation commerciale ou impression systématique est constitutive d’uneinfraction pénale. Toute copie ou impression de ce fichier doit conte-nir la présente mention de copyright.Article numérisé dans le cadre du programmeNumérisation de documents anciens mathématiqueshttp://www.numdam.org/163-On mean values of some zeta-functionsin the critical strippar ALEKSANDAR IVI0106Dedzcated to the memory of Robert RankinRÉSUMÉ. Un entier k ~ 3 et un réel 03C3 tel que ½  03C3  1 étantfixés, on considère dans la formule asymptotique$$le terme erreur R(k, 03C3; T), pour lequel nous montrons de nouvellesbornes lorsque min(03B2k, 03C3*k)  03C3  1. Nous obtenons également desmajorations nouvelles pour les termes erreur dans le développe-ment des moments d’ordre deux des fonctions zeta de formesparaboliques holomorphes et des séries de Rankin-Selberg.ABSTRACT. For a fixed integer k ~ 3, and fixed ½  03C3  1 weconsider$$where R(k, 03C3; T) = 0(T) (T~ ~) is the error term in the aboveasymptotic formula. Hitherto the sharpest bounds for R(k, 03C3; T)are derived in the range min(03B2k, 03C3*k)  03C3  1. We also obtainnew mean value results for the zeta-function of holomorphic cuspforms and the Rankin-Selberg series.1. IntroductionThe aim of this paper is to provide asymptotic formulas for the 2k-thmoment of the Riemann zeta-function ~(s) and some related Dirichlet seriesin the so-called "critical strip" 2  a = 9te s  1. For the zeta-functionour results are relevant when l~ &#x3E; 3 is a fixed integer, where henceforths = a + it will denote a complex variable. Mean values of (((s) on the"critical line" a = ~ behave differently (see e.g., [4]), while the problemManuscrit recu le 11 septembre 2001.164of mean values for 0  Q  ~ can be reduced to the range .1  Q  1 bymeans of the functional equation for ((s), namelyMean values of ((s) for -1  1 in the cases k = 1 and k = 2 have beenextensively studied, and represent one of the central themes in zeta-functiontheory. One has (see [10, Theorem 2])and (see [7, Theorem 2])which are the sharpest hitherto published asymptotic formulas valid in thewhole range .1  a  1. These results have been obtained by specialmethods, and cannot be generalized to higher moments. The formula forthe general 2k-th moment of ((s) can be conveniently written (cf. [4,Chapter 8]) aswhere 1  Q  1, T - oo, and the arithmetic function dk(n)denotes, as usual, the number of ways n may be written as a product ofk factors (so that dk(n) is generated by ~’’~(s), and d2(n) = d(n) is thenumber of divisors of n) . In [4, Chapter 8] it was proved thatwhere henceforth E denotes arbitrarily small constants, not necessarily thesame ones at each occurrence, and crk is the infimum of ~* ( &#x3E; ~ ) for whichholds for any given é. Writing further the bounds for R(k, d; T) as165and using the known bounds for when 3 ::; 1~  6, it follows from (1.4)that we haveAs indicated in [4], explicit values for ck (a) could be given for any fixedk &#x3E; 1, but the expressions in general would be cumbersome, so only explicitvalues were given for 2  k  6. The point of (1.3)-(1.5) lies in the factthat each value satisfies  1 (i.e., when (1.3) becomes a trueasymptotic formula), precisely for the range given in (1.5). However, as aapproaches 1, the values of become rather poor and they do not tendto zero, as one expects.The problem of mean values of a Dirichlet series F(s) (in this context2k-th moments of F(s) can be regarded simply as the mean square of(k E N)) can be treated in various degrees of generality. Here we shallmention only the classes of Dirichlet series treated by Chandrasekharan-Narasimhan (see [2], [3]), Perelli [14], Richert [17] and Selberg [18]. Re-cently S. Kanemitsu et al. obtained in [12] a mean value theorem for ageneral class of Dirichlet series possessing a functional equation with mul-tiple gamma-factors. The merit of their result, which is in part based onideas of Matsumoto [13], is a relatively good value of the exponent in theerror term as Q approaches the abscissa of absolute convergence of theDirichlet series in question. In particular, the result of [12] can be appliedto higher power moments of ~(s). In this case in the notation of [12] onehasTheir Theorem 4 gives then, in the notation of (1.3),for k &#x3E; 2 andWhen (1.6) is compared with (1.3)-(1.4) it transpires that it holds for apoorer range, but the exponent in the error term is much sharper as agrows, and it tends to 0 as a --7 1 - 0, as one expects.In what follows we may assume o-  1, since we have the asymptoticformula166which was proved in [1]. In (1.8) one can take l~ E C arbitrary, but fixed.Thus (1.8), obtained by a special method that cannot be adapted to therange or  1, yields a better error term than the one obtainable from anyof the previous bounds ( 1.1 )-t 1.7) .The plan of the paper is as follows. In Section 2 we shall formulate theresults (Theorem 1 and Theorem 2) concerning the higher moments of C(s),the proofs of which will be given in Section 3. In Section 4 we shall dealwith the mean value of the Rankin-Selberg series, and in Section 5 withthe mean values of the zeta-function of holomorphic modular forms and itssquare.2. Higher moments of the zeta-functionThe aim of this section is to furnish new bounds for R(k, a; T), whichwill improve both (1.4) and (1.6). We shall formulate now our results, withthe remark that Theorem 2 is based on the use of the defining property of~~ and it gives good bound for R(k, cr; T) when a is close to Theorem 1,on the other hand, is derived by using the values of the constant #k in themean square estimates for the divisor problem. Namely we let, as usual,where is the error term in the asymptotic formula for the summatoryfunction of dk(n) (cf. (3.1)). Theorem 1 will provide good results for valuesof Q close to 1. Results of similar type for the general case and the caseof the Rankin-Selberg series can be found in [12] and [13]. However in theproof of Theorem 1 we shall avoid using the Cauchy-Schwarz inequality andtherefore obtain a sharper value of the exponent than we would obtain byfollowing the ideas of [12] and [13].Theorem 1. For fcxed a satisfying  a  1 and every fixed3, we haveTheorem 2. For fixed a satisfying Q~  a  1 and every fixed integerk &#x3E; 3, we haveRemark 1. Note that (2.2) improves (1.6). Namely we havefor167But from (1.7) it follows thatforEquality in (2.4) holds only for k = 3, since 63 = 3 (see [4]). But we have/34 = 1 and #k  (k - 1)1(k + 2) for k &#x3E; 4 (see [17]), hence in (2.4) wehave strict inequality for k &#x3E; 3. This means that (2.2) improves both theexponent of the error term in (1.6), and at the same time it holds in a widerinterval than the one given by (1.7).We also note thatis equivalent towhich is obvious. This means that (2.3) of Theorem 2 improves (1.4) in thewhole range Qk  a  1.3. Proof of Theorem 1 and Theorem 2We write as usual, for k E N‘,where is a polynomial of degree k - 1, whose coefficients (whichdepend on k) may be explicitly evaluated. Using the Stieltjes integralrepresentation and (3.1) we have, for 1 « X W Tc (C &#x3E; 0), Q &#x3E; 1 andk &#x3E; 2 a fixed integer,where + P/ From the definition (2.1) of ,~~ it follows that,for any given Y » 1, there exists X E [Y, 2Y] such thatHenceforth we assume that X is chosen in such a way that it satisfies,besides 1 « X « Tc, also the bound in (3.3). Repeated integration by168parts yields ’which provides analytic continuation of the left-hand side of (3.4) to C. Wealso haveNote that the last integral converges absolutely for a &#x3E; in view ofthe Cauchy-Schwarz inequality for integrals and the definition (2.1) of Therefore from (3.1)-(3.5) we obtain, for  Q  1 and T  t 2T,Observe now that (2.2) follows fromon replacing T by T2-i (j E N) and summing all the results. To evaluatethe integral in (3.7), we suppose that  1, we use (3.6) andThe reason of this splitting of the sum in two sums is to have rn and n differby unity at least, which is expedient to have in the integration that will169follow. Now note that we have, by the mean value theorem for Dirichletpolynomials (see [4, Chapter 4]) and dk(n) « 7~BTo evaluate the mean square of ibi we may proceed directly by squaringout the modulus, or we may use Lemma 4 of [8], which says thatholds if g(x) is a real-valued, integrable function on ~a, /3~, a subinterval of[2, oo), which is not necessarily finite. We shall obtainWe haveBy direct integration it is found that170on using the elementary inequalitySimilarly, by using the first derivative test (see [4, Lemma 2.1]), we obtainwhere the interchange of the order of integration is justified by absoluteconvergence. Therefore from (3.10)-(3.12) it follows thatso that finally from (3.8)-(3.10) and (3.13) we obtainNow in (3.14) we set X ~-2~ ~ ~’X 1 +~~ ~~~, namelywhere the constant c &#x3E; 0 is chosen in such a way that (3.3) is satisfied.With the choice (3.15) it is seen that (3.14) becomes (3.7), and the proofof Theorem 1 is completed. 0Corollary 1.171The formulas follow from Theorem 1 with the values /3s = 3, ~4 = -/?6  2 (see [4]) and #5  o (see [20]).Remark 2. It transpires that the Lindelof hypothesis (~(Q+it) G Itl£ fora &#x3E; ~) is equivalent toNamely the Lindelof hypothesis implies that J3k = (k - 1)/(2k) for k &#x3E; 2(see [4, Chapter 13]), in which case (3.16) follows from (2.2) and Theorem 1.Conversely, if (3.16) holds, then by [4, Lemma 7.1] we have, for T~  H 1/2 Twhich yields the Lindelof hypothesis on taking H = ~T and letting 1~ -~ oo.Remark 3. Other explicit results can be obtained from Theorem 1 withthe bounds for ~3~ furnished by [9], some of which are hitherto the sharpestones. Our method of proof can be used to obtain a sharpening of thegeneral result proved in [12], since we did not use the Cauchy-Schwarzinequality in estimating ab dt, which was done in [12] and [13]. Namelywe integrated directly the expressions in question, which led to a sharperestimate than the one that would have resulted from the application of theCauchy-Schwarz inequality.Proof of Theorem 2. From the well-known Mellin inversion integralwe obtainWe move the line ofintegration in (3.17) to 9ie w = or* - a-. In doing this we encounter the polew = 1 - s with residue O(T-A) for any fixed A &#x3E; 0 in view of Stirling’s172formula for the gamma-function. There is also the simple pole at w = 0with residue ~’k(s). Therefore from (3.17) it follows thatwhereConsequently we have, by the mean value theorem for Dirichlet polynomi-als,We also have, by the definition of or*on taking A sufhciently large. Finally by using the Cauchy-Schwarz in-equality we obtainPutting together all the estimates, replacing T by T2-i (j &#x3E; 1) and sum-ming over j we obtainNow we take173to obtainIOn noting that the conditionreduces to a, which is certainly true, we obtain then (2.3) .Corollary 2.The above formulas follow from Theorem 2 with the values (see [4, Chap-ter 8]) ~3  y~, o-4  ~ , ~~  so , a6  ? . They improve (1.5) and comple-ment those furnished by Corollary 1.4. The mean value of the Rankin-Selberg seriesThe arguments used in the proof of Theorem 1 and Theorem 2 are of ageneral nature and can be adapted to obtain mean value results for a wideclass of Dirichlet series. Instead of working out the details in the generalcase, which would entail various technicalities, we prefer to conclude byconsidering two specific examples. In this section we shall deal with themean value of the so-called Rankin-Selberg series (see R.A. Rankin [15],[16])and in Section 5 we shall consider the zeta-function attached to holomorphiccusp forms. Here as usual a(n) denotes the n-th Fourier coefficients of aholomorphic cusp form cp(z) of weight /-. with respect to the full modulargroup SL(2, Z). We also suppose that is a normalized eigenfunction174for the Hecke operators T(n), so that a(1) = 1 and a(n) E R We have (seeP-’" r- , - - , , -with Rankin’s classical estimate (see andThis means that analogously to (3.6) we havefor ~  1, T ~ 2T, where 1 « X  T~ and X (E [K, 2V]) satisfies(this is the analogue of (3.3))Then we writewithand considerSimilarly as in the proof of Theorem 1 we find thatWith the choice X = bT2, where b &#x3E; 0 is a suitable constant, we obtain175which easily gives thenTheorem 3. For fixed a satisfying 1  a  1 we haveRemark 4. The asymptotic formula (4.2) improves, for 1  a  1, theresult of K. Matsumoto [13] who provedwithFor -1  Q  4 our result is slightly weaker than the corresponding resultof [13], namelybut it should be remarked that (4.2) is a true asymptotic formula only inthe range 1  Q  1.5. The mean value of the zeta-function of cusp formsWe retain the notation of Section 4 and consider (see [6]) the Dirichletserieswhich may be continued analytically to an entire function over C. In (5.1)the arithmetic functionis the "normalized" function of cusp form coefficients. This function is"small" , since it satisfies 3(n) « d(n) by Deligne’s classical estimate. Weshall also considerwhere176is the convolution of a(n) with itself. The mean values of F(s) and F2(s)were considered in [6]. It was proved there that, for a fixed,andwithandNote that from (5.5) and (5.7) it transpires that we can obtain a trueasymptotic formula for the fourth moment of F(a+it) for i  a  1. Thisreflects the fact that we have (see [6])only for Q 2 ~, and any improvement of the range for which (5.8) holdswould result in the improvement of the bound for K(a°;T). We shall im-prove on (5.6) and (5.7) by provingTheorem 4. If H(Q;T) is defined by (5.4), then for a fixed we haveTheorem 5. If K(a; T) is defined by (5.5), then for cr fixed we haveProof of Theorem 4 and Theorem 5. The first bounds in (5.9) and(5.10) are the analogues of (2.3) of Theorem 2 corresponding to the values~i = 2 and ~2 = 8, which follow from (5.4) and (5.8), respectively. Themethod of proof of Theorem 2 may be used, since177Similarly the second bounds in (5.9) and (5.10) are the analogues of (2.2)of Theorem 1 corresponding to the values #i = 4 and #2 = 1, respectively.Namely if we defineandthen p = 4 (this corresponds to fi2 = ~ in the classical Dirichlet divisorproblem) (see [6]; this corresponds to,84 = $ in the Dirichlet divi-sor problem for ~4(x)). Thus proceeding as in the proof of Theorem 1 andkeeping in mind again that (5.11) holds, we shall obtain the second boundsin (5.9) and (5.10). Clearly the bounds in (5.9) and (5.10) improve (5.6) and(5.7), respectively. Although the first bound in (5.10) holds for 2  ~  4,it is relevant only in the range Q &#x3E; i, when 16 ( 1 - cr)/(ll 2013 8Q)  1, when(5.5) becomes a true asymptotic formula.References[1] R. BALASUBRAMANIAN, A. IVI0106, K. RAMACHANDRA, An application of the Hooley-Huxleycontour. Acta Arith. 65 (1993), 45-51.[2] K. CHANDRASEKHARAN, R. NARASIMHAN, Functional equations with multiple gamma factorsand the average order of arithmetical functions. Ann. of Math. 76 (1962), 93-136.[3] K. CHANDRASEKHARAN, R. NARASIMHAN, The approximate functional equation for a classof zeta-functions. Math. Ann. 152 (1963), 30-64.[4] A. IVI0106, The Riemann zeta-function, John Wiley &#x26; Sons, New York, 1985.[5] A. IVI0106, Large values of certain number-theoretic error terms. Acta Arith. 56 (1990), 135-159.[6] A. IVI0106, On zeta-functions associated with Fourier coefficients of cusp forms. Proceedingsof the Amalfi Conference on Analytic Number Theory (Amalfi, September 1989), Universitàdi Salerno, Salerno 1992, pp. 231-246.[7] A. IVI0106, Some problems on mean values of the Riemann zeta-function. J. Théor. NombresBordeaux 8 (1996), 101-123.[8] A. IVI0106, On some conjectures and results for the Riemann zeta-function and Hecke series.Acta Arith. 99 (2001), no. 2, 115-145.[9] A. IVI0106, M. OUELLET, Some new estimates in the Dirichlet divisor problem. Acta Arith. 52(1989), 241-253.[10] A. IVI0106, K. MATSUMOTO, On the error term in the mean square formula for the Riemannzeta-function in the critical strip. Monatsh. Math. 121 (1996), 213-229.[11] A. IVI0106, K. MATSUMOTO, Y. TANIGAWA, On Riesz means of the coefficients of the Rankin-Selberg series. Math. Proc. Camb. Phil. Soc. 127 (1999), 117-131.[12] S. KANEMITSU, A. SANKARANARAYANAN, Y. TANIGAWA, A mean value theorm for Dirichletseries and a general divisor problem, to appear.[13] K. MATSUMOTO, The mean values and the universality of Rankin-Selberg L-functions. Proc.Turku Symposium on Number Theory in Memory of K. Inkeri, May 31-June 4, 1999, Walterde Gruyter, Berlin etc., 2001, in print.[14] A. PERELLI, General L-functions. Ann. Mat. Pura Appl. 130 (1982), 287-306.178[15] R. A. RANKIN, Contributions to the theory of Ramanujan’s function 03C4(n) and similar arith-metical functions. II, The order of Fourier coefficients of integral modular forms. Math.Proc. Cambridge Phil. Soc. 35 (1939), 357-372.[16] R. A. RANKIN, Modular Forms. Ellis Horwood Ltd., Chichester, England, 1984.[17] H.-E. RICHERT, Über Dirichletreihen mit Funktionalgleichung. Publs. Inst. Math. (Belgrade)11 (1957), 73-124.[18] A. SELBERG, Old and new conjectures and results about a class of Dirichlet series. Proc.Amalfi Conf. Analytic Number Theory, eds. E. Bombieri et al., Università di Salerno,Salerno, 1992, pp. 367-385.[19] E. C. TITCHMARSH, The theory of the Riemann zeta-function, 2nd edition.Oxford UniversityPress, Oxford, 1986.[20] ZHANG WENPENG, On the divisor problem. Kexue Tongbao 33 (1988), 1484-1485.Aleksandar Ivi6Katedra Matematike RGF-aUniversiteta u BeograduÐu0161ina 711000 BeogradSerbia (Yugoslavia)aivic8matf.bg.ac.yu, aivic0rgf.bg.ac.yu

On mean values of some zeta-functions in the critical strip

Abstract

Similar works

Full text

Available Versions

Journal de Théorie des Nombres de Bordeaux

Numérisation de Documents Anciens Mathématiques

Crossref