LOWER BOUNDS FOR MOMENTS OF ζ (ρ) 1. Introduction

Transcription

1 LOWER BOUNDS FOR MOMENTS OF ζ ρ MICAH B. MILINOVICH AND NATHAN NG Abstract. Assuig the Riea Hypothesis, we establish lower bouds for oets of the derivative of the Riea zeta-fuctio averaged over the otrivial zeros of ζs. Our proof is based upo a recet ethod of Rudick ad Soudararaja that provides aalogous bouds for oets of L-fuctios at the cetral poit, averaged over failies.. Itroductio Let ζs deote the Riea zeta-fuctio. I this article we are iterested i obtaiig lower bouds for oets of the for J k T = ζ ρ 2k NT where k N ad the su rus over the o-trivial coplex zeros ρ = β + iγ of ζs. As usual, we let the fuctio NT = = T 2π log T 2π T + Olog T 2 2π deote the uber of zeros of ζs up to a height T couted with ultiplicity. Idepedetly, Goek [3] ad Hejhal [5] have cojectured that J k T log T kk+2 for each k R. By odelig the Riea zeta-fuctio ad its derivative usig characteristic polyoials of rado atrices, Hughes, Keatig, ad O Coell [6] have refied this cojecture to state that J k T C k log T kk+2 for a precise costat C k whe k C ad Rek > 3/2. However, we o loger believe this cojecture to be true for Rek < 3/2. This is sice we expect there exist ifiitely ay zeros ρ such that ζ ρ γ /3 ε for each ε > 0. Results of the sort suggested by these cojectures are oly kow for a few sall values of k. See, for istace, the results of Goek [] for the case k = ad Ng [8] for the case k = 2. Also, Goek [3] obtaied a lower boud i the case k =. Our ai result is to obtai a lower boud for J k T for each k N of the order of agitude that is suggested by these cojectures. Theore. Assue the Riea Hypothesis ad let k N. The for sufficietly large T we have ζ ρ 2k k log T kk+2. NT Jue 5, Matheatics Subject Classificatio M06, M26. The first author is partially supported by the Natioal Sciece Foudatio FRG grat DMS The secod author is supported by a NSERC research grat.

2 2 MICAH B. MILINOVICH AND NATHAN NG Uder the assuptio of the Riea Hypothesis, Miliovich [7] has recetly show that J k T k,ε log T kk+2+ε for k N ad ε > 0 arbitrary. Whe cobied with Theore, this result leds strog support for the cojecture of Goek ad Hejhal for k a positive iteger. Theore ca be used to exhibit large values of ζ ρ. iediate corollary we have the followig result. For exaple, as a Corollary.. Assue the Riea Hypothesis ad let ρ = 2 o-trivial zero of ζs. The for each A > 0 the iequality ζ ρ log γ A + iγ deote a 3 is satisfied ifiitely ofte. This result was previously prove by Ng [0] by a applicatio of Soudararaja s resoace ethod [3]. The preset proof is sipler ad provides ay ore zeros ρ such that 3 is true. O the other had, the resoace ethod is capable of detectig uch larger values of ζ ρ assuig a very weak for of the geeralized Riea hypothesis. Our proof of Theore relies o cobiig a ethod of Rudick ad Soudararaja [, 2] with a ea-value theore of Ng our Lea 2 ad a well-kow lea of Goek our Lea 3. It is likely that our proof ca be adapted to prove a lower boud for J k T of the cojectured order of agitude for all ratioal k with k i a aer aalogous to that suggested i []. Let k N ad defie, for ξ, the fuctio A ξ s = ξ s. Assuig the Riea Hypothesis, we will estiate Σ = ζ ρa ξ ρ k A ξ ρ k ad Σ 2 = Aξ ρ 2k where the sus ru over the o-trivial zeros ρ = 2 +iγ of ζs. Hölder s iequality iplies that ζ ρ 2k Σ 2k Σ2 2k, ad so we see that Theore will follow fro the estiates Σ T log T k2 +2 ad Σ 2 T log T k It is coveiet to express Σ ad Σ 2 slightly differetly. Assuig the Riea Hypothesis, ρ = ρ for ay o-trivial zero ρ of ζs. Thus, A ξ ρ = A ξ ρ. This allows us to re-write the sus i as Σ = ζ ρa ξ ρ k A ξ ρ k ad Σ 2 = A ξ ρ k A ξ ρ k. 5 It is with these represetatios of Σ ad Σ 2 that we establish the bouds i 4.

3 LOWER BOUNDS FOR MOMENTS OF ζ ρ 3 2. Soe preliiary estiates For each real uber ξ ad each k N, we defie the arithetic sequece of real ubers τ k ; ξ by τ k ; ξ k s = = Aξ s s k. 6 ξ k ξ The fuctio τ k ; ξ is a trucated approxiatio to the arithetic fuctio τ k the k-th iterated divisor fuctio which is defied by ζ k s = = k = s = τ k s 7 for Res >. We require a few estiates for sus ivolvig the fuctios τ k ad τ k ; ξ i order to establish the bouds for Σ ad Σ 2 i 4. We use repeatedly that, for x 3 ad k, l N, x τ k τ l k,l log x kl 8 where the iplied costats deped o k ad l. These bouds are well-kow. Fro 6 ad 7 we otice that τ k ; ξ is o-egative ad τ k ; ξ τ k with equality holdig whe ξ. I particular, choosig k = l i 8 we fid that, for ξ 3, τ log ξ k2 k 2 k τ k ; ξ2 τ k 2 k log ξ k2. 9 ξ ξ k ξ k 3. A Lower Boud for Σ I order to establish a lower boud for Σ, we require a ea-value estiate for sus of the for SX, Y ; T = ζ ρxρy ρ where Xs = N x s ad Y s = y s N are Dirichlet polyoials. For Xs ad Y s satisfyig certai reasoable coditios, a geeral forula for SX, Y ; T has bee established by the secod author [9]. Before statig the forula, we first itroduce soe otatio. For T large, we let L = log T 2π ad N = T ϑ for soe fixed ϑ 0. The fuctios µ ad Λ are used to deote the usual arithetic fuctios of Möbius ad vo Magoldt. Also, we defie the arithetic fuctio Λ 2 by Λ 2 = µ log 2 for each N.

4 4 MICAH B. MILINOVICH AND NATHAN NG Lea 2. Let x ad y satisfy x, y τ l for soe l N ad assue that 0 < ϑ < /2. The for ay A > 0, ay ε > 0, ad sufficietly large T we have SX, Y ; T = T x y P 2 L 2P L log + Λ log 2π N T 4π N y x Q 2L log + T 2π + O A T log T A + T 3/4+ϑ/2+ε a,b N a,b= ra; b ab g i N a, N b where P, P 2, ad Q 2 are oic polyoials of degrees,2, ad 2, respectively, ad for a, b N the fuctio ra; b satisfies the boud ra; b Λ 2 a + log T Λa. 0 y ag x bg g Proof. This is a special case of Theore.3 of Ng [9]. Lettig ξ = T /4k, we fid that the choices Xs = A ξ s k ad Y s = A ξ s k satisfy the coditios of Lea 2 with ϑ = /4, N = ξ k, x = τ k ; ξ, ad y = τ k ; ξ. Cosequetly, for this choice of ξ, Σ = T 2π ξ k ξ k τ k ; ξτ k ; ξ P 2 L 2P L log + Λ log T τ k ; ξτ k ; ξ Q 2 L log 4π ξ k + T ra; b τ k ag; ξτ k bg; ξ + O T 2π ab g a,b ξ k a,b= = S + S 2 + S 3 + OT, g i N a, N b say. To estiate S, otice that, for T sufficietly large, ξ k = T /4 iplies that P 2 L 2P L log + Λ log L 2 ad oreover, by 9, ξ k ξ k τ k ; ξτ k ; ξ τ k ; ξ2 log T k2. ξ k Thus, S T log T k2 +2. Sice Q 2 L log L 2, we ca boud S 2 by usig the iequalities τ k ; ξ τ k ad τ k τ k τ k. I particular, by twice usig 8, we fid that S 2 T L 2 τ k τ k τ k T L 2 ξ k T T log T 2+kk +k T log T k2 +. τ k τ k T τ k

5 LOWER BOUNDS FOR MOMENTS OF ζ ρ 5 It reais to cosider the cotributio fro S 3. Agai usig the iequalities τ k ; ξ τ k ad τ k τ k τ k alog with 0, it follows that S 3 is bouded by Λ 2 a + log T Λa τ k aτ k gτ k bτ k g ab g a,b ξ k g ξ k a T Λ 2 a + log T Λaτ k a a b T log T 2+k +kk = log T k2 +. τ k b b g T τ k gτ k g g Cobiig this with our estiates for S ad S 2, we coclude that Σ T log T k A Upper Boud for Σ 2 Assuig the Riea Hypothesis, we iterchage the sus i 5 ad fid that Σ 2 = NT τ k ; ξ2 + 2Re ξ k ξ k τ k ; ξτ k ; ξ < ξ k ρ where NT deotes the uber of o-trivial zeros of ζs up to a height T. Recallig that ξ = T /4k ad usig 2 ad 9, it follows that NT τ k ; ξ2 ξ k T log T k I order to boud the secod su o the right-had side of, we require the followig versio of the Ladau-Goek explicit forula. Lea 3. Let x, T > ad let ρ = β + iγ deote a o-trivial zero of ζs. The x ρ = T 2π Λx + O x log2xt log log3x + O log x i T, x + O log2t i T, x log x where x deotes the distace fro x to the closest prie power other tha x itself ad Λx = log p if x is a positive itegral power of a prie p ad Λx = 0 otherwise. Proof. This is a result of Goek [2, 4].

6 6 MICAH B. MILINOVICH AND NATHAN NG Applyig the lea, we fid that ξ k τ k ; ξτ k ; ξ < ξ k ρ T τ k ; ξτ k ; ξλ = 2π ξ k < ξ k + O L log L τ k ; ξτ k ; ξ ξ k < ξ k + O τ k ; ξτ k ; ξ log ξ k < ξ k + O log T τ k ; ξτ k ; ξ log ξ k < ξ k = S 2 + S 22 + S 23 + S 24, say. Sice we oly require a upper boud for Σ 2 which, by defiitio, is clearly positive, we ca igore the cotributio fro S 2 because all the o-zero ters i the su are egative. I what follows, we use ε to deote a sall positive costat which ay be differet at each occurrece. To estiate S 22, we ote that τ k ; ξ τ k ε ε which iplies S 22 T /4+ε. Turig to S 23, we write as q + l with 2 < l 2 ad fid that S 23 T ɛ ξ k q ξk + 2 <l 2 q + l where x deotes the greatest iteger less tha or equal to x. Notice that q+ l = l if q is a prie power ad l 0, otherwise q + l is 2. Hece, S 23 T ε ξ k T ε q ξk + l 2 Λq 0 ξ k q ξk + T /4+ɛ. l + ξ k q ξk + l 2 It reais to cosider S 24. For itegers < ξ k, let = + l. The Cosequetly, S 24 T ɛ log = log l ξ k > l. + l l T ɛ ξ k = T /4+ɛ. 3 l ξ k Cobiig 2 with our estiates for S 22, S 23, ad S 24 we deduce that Σ 2 T log T k2 + which, whe cobied with our estiate for Σ, copletes the proof of Theore.

7 LOWER BOUNDS FOR MOMENTS OF ζ ρ 7 Refereces [] S. M. Goek, Mea values of the Riea zeta-fuctio ad its derivatives, Ivet. Math , [2] S. M. Goek, A forula of Ladau ad ea values of ζs i Topics i Aalytic Nuber Theory, S. W. Graha ad J. D. Vaaler, eds., Uiv. Texas Press, Austi, Tex., 985, [3] S. M. Goek, O egative oets of the Riea zeta-fuctio, Matheatika , [4] S. M. Goek, A explicit forula of Ladau ad its applicatios to the theory of the zeta fuctio, Cotep. Math , [5] D. Hejhal, O the distributio of log ζ /2 + it, i Nuber Theory, Trace Forulas, ad Discrete Groups, K. E. Aubert, E. Bobieri, ad D. M. Goldfeld, eds., Proceedigs of the 987 Selberg Syposiu, Acadeic Press, 989, [6] C. P. Hughes, J. P. Keatig, ad N. O Coell, Rado atrix theory ad the derivative of the Riea zeta-fuctio, Proc. Roy. Soc. Lodo A , [7] M. B. Miliovich, Upper bouds for oets of ζ ρ, preprit. [8] N. Ng, The fourth oet of ζ ρ, Duke Math J [9] N. Ng, A discrete ea value of the derivative of the Riea zeta fuctio, preprit. [0] N. Ng, Extree values of ζ ρ, preprit. [] Z. Rudick ad K. Soudararaja, Lower bouds for oets of L-fuctios, Proc. Natl. Sci. Acad. USA , [2] Z. Rudick ad K. Soudararaja, Lower bouds for oets of L-fuctios: syplectic ad orthogoal exaples, i Multiple Dirichlet Series, Autoorphic Fors, ad Aalytic Nuber Theory, Friedberg, Bup, Goldfeld, ad Hoffstei, eds., Proc. Syp. Pure Math., vol. 75, Aer. Math. Soc., [3] K. Soudararaja, Extree values of L-fuctios, preprit. Micah B. Miliovich Math Departet Uiversity of Rochester Rochester, NY 4627 USA icah@ath.rochester.edu Natha Ng Departet of Matheatics ad Statistics Uiversity of Ottawa 585 Kig Edward Aveue Ottawa, ON KN 6N5 Caada g@uottawa.ca