A NOTE ON S(t) AND THE ZEROS OF THE RIEMANN ZETA-FUNCTION

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1 Bull. London Mah. Soc C 2007 London Mahemaical Sociey doi: /blms/bdm032 A NOTE ON S AND THE ZEROS OF THE RIEMANN ZETA-FUNCTION D. A. GOLDSTON and S. M. GONEK Absrac Le πs denoe he argumen of he Riemann zea-funcion a he poin 1/2 + i. Assuming he Riemann hypohesis, we sharpen he consan in he bes currenly known bounds for S and for he change of S in inervals. We hen deduce esimaes for he larges mulipliciy of a zero of he zea-funcion, and for he larges gap beween he zeros. 1. Inroducion We assume he Riemann hypohesis RH hroughou his paper. Le N denoe he number of zeros ρ =1/2+iγ of he Riemann zea-funcion wih ordinaes in he inerval 0,]. Then, for 2, N = log S+O, 1.1 where, if is no he ordinae of a zero, S denoes he value of 1/π arg ζ1/2+i obained by coninuous variaion along he sraigh line segmens joining 2, 2 + i, and 1/2 + i, saring wih he value 0 see [10]. If is he ordinae of a zero, we se S = 1 2 lim ε 0 +{S + ε+s ε}. I follows from 1.1 ha N + h N = h log 1 + h 2 + S + h S+O 1.2 for 0 <h. Lilewood [5] proved, assuming he Riemann hypohesis, ha S log log log, 1.3 where here he noaion f g means he same as f = Og. Hence he number of zeros wih ordinaes in an inerval, + h] saisfies N + h N h log log log log, 1.4 provided ha 0 <h,say. The bounds in 1.3 and 1.4 have no been improved over he las eighy years, excep in he size of he implied consans. Our goal in his paper is o sharpen hese resuls. Received 15 November 2005; revised 24 Augus 2006; published online 4 May Mahemaics Subjec Classificaion 11M26. The research of boh auhors was suppored in par by a Naional Science Foundaion FRG gran DMS The firs auhor was also parially suppored by NSF gran DMS , and he second auhor by NSF gran DMS The auhors wish o hank he Isaac Newon Insiue for is hospialiy during heir work on his paper, and also he American Insiue of Mahemaics.

2 A NOTE ON S AND THE ZEROS OF THE RIEMANN ZETA-FUNCTION 483 Theorem 1. we have Assume he Riemann hypohesis. Le be large, and le 0 <h. Then N + h N h log 2 + o1 log log log. 1.5 In ligh of 1.2, his is equivalen o saying ha 1 log S + h S 2 + o1 log log for 0 <h. Using his, we obain he following heorem. 1.6 Theorem 2. Assume he Riemann hypohesis. Then for sufficienly large we have log S 2 + o1 log log. 1.7 To deduce Theorem 2 from Theorem 1, we use he uncondiional esimae of Lilewood [5], ha T Su du log T, which implies ha 0 +log 2 Su du log. Therefore, for sufficienly large, here is an h wih 0 h log 2 such ha S + h 1. Rewriing 1.6 as log log 2 + o1 log log + S + h S 2 + o1 + S + h, 1.8 log log we obain he upper bound for S from he righ-hand inequaliy. We obain he lower bound by using an h for which S + h 1, ogeher wih he lef-hand inequaliy. The following is an almos immediae corollary of Theorem 1. Corollary 1. Assume he Riemann hypohesis. Le mγ denoe he mulipliciy of he zero 1/2+iγ. Then if γ is sufficienly large, we have log γ mγ 2 + o1 log log γ. 1.9 Moreover, if γ and γ are consecuive ordinaes and γ<γ, hen γ π γ 1 + o log log γ To deduce 1.9, ake = γ h/2 in1.5 wih h = o1/ log log γ. To deduce 1.10, assume ha N + h N =0in1.5, and solve for h. There has been some earlier work on Theorem 2. In place of he consan 1/2, Ramachandra and Sankaranarayanan [8] obained in 1993, and Fujii [2] obained.67 in For a recen survey of his area, see [4]. I is ineresing ha Brumer [1] obained he same consan 1/2 for a similar bound for he rank of an ellipic curve in erms of he conducor. 2. Proof of Theorem 1 We begin by saing wo lemmas. The firs is a form of he Guinand Weil explici formula.

3 484 D. A. GOLDSTON AND S. M. GONEK Lemma 1. Le hs be analyic in he srip Im s 1/2+ε for some ε>0, and assume ha hs 1 + s 1+δ for some δ>0when Re s. Le hw be real-valued for real w, and se ĥx = hwe ixw dw. Then ρ 1/2 h = h + h i 2i ĥ0 log π + hu Re Γ 1 2i ρ Γ 4 + iu du 2 1 Λn log n log n ĥ n ĥ, n=2 where Γ /Γ is he logarihmic derivaive of he gamma funcion, and Λn is he von Mangold funcion defined o be log p if n = p m, p a prime and m 1, and zero oherwise. This is a specializaion of [3, Theorem 5.12], and in paricular [3, equaion 25.10]. The condiions in [3] are ha ĥ is an infiniely differeniable funcion wih compac suppor, which will be saisfied in our applicaion below; however, i is also no hard o prove he lemma wih he condiions ha we have saed. Lemma 2. Le L and δ be posiive real numbers, and le w = u + iv. There exis even enire funcions F + w and F w wih he following properies: i F u χ [ L,L] u F + u for all real u; ii F +u du 2L +1/δ, and F u du 2L 1/δ; iii F ± w e δ Im w ; iv F ± u min1, δ 2 u L 2 for u >L; v ˆF ± x =0for x δ; vi ˆF ± x = sin Lx/πx + O1/δ. This is essenially [7, Lemma 2]. Funcions of his ype were consruced by A. Selberg, who gives a nice discussion of hem in [9]. For a proof of his lemma, see H. L. Mongomery [6] and J. D. Vaaler [11]. The slighly less familiar propery iv is obained from [11, Lemma 5]. To prove Theorem 1, we use Lemma 1 wih hw =F w, where is large and posiive and F denoes eiher he funcion F + or F from Lemma 2. We assume ha he parameers δ and L implici in he definiion of F ± saisfy he condiions δ 1 and 0 <L Clearly, ĥx =e ix ˆF x. Therefore, by Lemma 2i and Lemma 2ii, or Lemma 2vi, we have ĥ0 = ˆF 0=2L + O, δ and log n ĥ = n i ˆF We also see ha by Lemma 2iii. We will now show ha 1 log n, ĥ log n = n i ˆF h + h 2i 1 = F 2i 2i + F 2i e πδ F u Re Γ Γ 4 + iu 2 log n. du = 1 log ˆF 0 + O

4 Firs, since A NOTE ON S AND THE ZEROS OF THE RIEMANN ZETA-FUNCTION 485 Re Γ Γ 4 + iu 2 we see by Lemma 2iv and 2.2 ha +4 F u Re Γ Γ 4 + iu 2 log u +2, du +4 log, logu +2 δ 2 u 2 2 du and similarly for he inegral over, 4 ]. Nex, by Sirling s formula for large, ogeher wih Lemma 2ii and he previous argumen using Lemma 2iv, we have +4 4 F u Re Γ Γ iu +4 2 du = 4 F u log u O 1+u 2 du +4 = 4 F u log O du = log log/2 + L F u du + O 2 = log 2 ˆF 0 + O 1. On combining hese esimaes, 2.3 follows. Insering hese resuls ino 2.1, we obain F γ = ˆF 0 log + Oe πδ 1 Λn log n log n n i ˆF + n i ˆF. γ n=2 n 2.4 By Lemma 2v and Lemma 2vi, he sum on he righ is Λn sinl log n cos log n + O 1 n e πδ, n log n δ n<e δ n e δ where he las sum was esimaed rivially. Hence F γ = ˆF 0 log + Oe πδ. 2.5 γ Taking F o be F + and using Lemma 2i and Lemma 2ii, we find ha N + L N L 1 log 2L Oe πδ. δ We now ake πδ = log log 2 log log log and obain N + L N L L log π log 2 + o1 log log. Had we used F in 2.5 insead of F +, we would have found ha N + L N L L π log 1 log 2 + o1 log log. Combining hese wo inequaliies, we conclude ha N + L N L L π log 1 log 2 + o1 log log. Finally, replacing by + h/2 and aking L = h/2, we obain Theorem 1.

5 486 A NOTE ON S AND THE ZEROS OF THE RIEMANN ZETA-FUNCTION References 1. A. Brumer, The average rank of ellipic curves. I, Inven. Mah A. Fujii, An explici esimae in he heory of he disribuion of he zeros of he Riemann zea funcion, Commen. Mah. Univ. S. Paul H. Iwaniec and E. Kowalski, Analyic number heory, Amer. Mah. Soc. Colloq. Publ. 53 American Mahemaical Sociey, Providence, RI, A. A. Karasuba and M. A. Korolëv, The argumen of he Riemann zea funcion, Russian Mah. Surveys no. 3, in English, Uspekhi Ma. Nauk no. 3, in Russian. 5. J. E. Lilewood, On he zeros of he Riemann zea-funcion, Proc. Camb. Philos. Soc H. L. Mongomery, The analyic principle of he large sieve, Bull. Amer. Mah. Soc H. L. Mongomery and A. M. Odlyzko, Gaps beween zeros of he zea funcion, Topics in classical number heory, Vols I, II, Budapes, 1981, Colloq. Mah. Soc. Ja nos Bolyai 34 Norh-Holland, Amserdam, K. Ramachandra and A. Sankaranarayanan, On some heorems of Lilewood and Selberg. I, J. Number Theory A. Selberg, wih a foreword by K. Chandrasekharan, Colleced papers, Vol. II Springer, Berlin, E. C. Tichmarsh, edied and wih a preface by D. R. Heah-Brown, The heory of he Riemann zeafuncion, 2nd edn The Clarendon Press/Oxford Universiy Press, New York, J. D. Vaaler, Some exremal funcions in Fourier analysis, Bull. Amer. Mah. Soc. N.S D. A. Goldson Deparmen of Mahemaics San Jose Sae Universiy San Jose, CA USA goldson@mah.sjsu.edu S. M. Gonek Deparmen of Mahemaics Universiy of Rocheser Rocheser, NY USA gonek@mah.rocheser.edu