The Zeros of the Derivative of the Riemann Zeta Function Near the Critical Line

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1 Ki, H 008 The Zeros of the Derivative of the Riemann Zeta Function Near the Critical Line, International Mathematics Research Notices, Vol 008, Article ID rnn064, 3 pages doi:0093/imrn/rnn064 The Zeros of the Derivative of the Riemann Zeta Function Near the Critical Line Haseo Ki Department of Mathematics, Yonsei University, Seoul 0-749, Korea Correspondence to be sent to: haseo@yonseiackr We study the horizontal distribution of zeros of s which are denoted as ρ = β + iγ We assume the Riemann hypothesis which implies β / for any nonreal zero ρ, equality being possible only at a multiple zero of ζ s In this paper, we prove that lim inf β / log γ 0 if and only if, for any c > 0ands = σ + it with 0 < σ / < c/ log t t > t 0 c, we have ζ s = + Olog t, s ρ where ρ = / + iγ is the zero of ζ closest to s and to the origin, if there are two such We also show that if lim inf β / log γ 0, then for any c > 0ands = σ + it t > t c, we have log t σ log ζ s = O uniformly for / + c/ log t σ σ < Received June, 007; Revised June, 007; Accepted May 0, 008 Communicated by Prof Dennis Hejhal C The Author 008 Published by Oxford University Press All rights reserved For permissions, please journalspermissions@oxfordjournalsorg

2 H Ki Introduction The Riemann hypothesis RH states that the real part of any nonreal zero of the Riemann zeta function ζ s is/ A Speiser [4] has a theorem which says that RH is equivalent to the nonexistence of nonreal zeros of s inres < / We let ρ = β + iγ denote a zero of s where a sum over ρ is repeated according to multiplicity N Levinson and H L Montgomery [9] proved that for nonreal zeros of s, the average value of β with 0 <γ T is / + log log T/ log T But it is likely that β / is usually of the order / log T rather than log log T/ log T By a result of B C Berndt [], the number of zeros with ordinates less than T is 0<γ T = T π log T + Olog T 4πe Under RH, K Soundararajan [3] demonstrated the presence of a positive proportion of zeros of s in the region σ</ + ν/ log T for all ν 6 A positive proportion of zeros means lim inf T T π log T #{ρ : β / + ν/ log T,0<γ T} > 0 We introduce further results related to the theorem of K Soundararajan We arrange the zeros of the Riemann zeta function ζ s in the upper half-plane as ρ, ρ, with ρ n = β n + iγ n and 0 <γ γ, where a zero of multiplicity m appears precisely m times in this sequence RH is that β n = / for any n =,, 3, If two zeros ρ n ρ n have the same imaginary part, then n < n implies β n <β n Fora > 0, define D a = lim inf T T π log T # { n : γ n T, γ n+ γ n < a } log T In his remarkable work, Y Zhang [6] showed that not only unconditionally does there exist some ν>0 such that a positive proportion of zeros of s are in the region σ / <ν/log T, but also assuming RH and D a > 0 for any a > 0, that ν can be

3 The Zeros of the Derivative of the Riemann Zeta Function 3 arbitrary small Recently, Feng [4] proved Zhang s conditional result assuming only the same hypothesis for D a Thus it is very probable that lim inf β / log γ = 0 Assuming the truth of RH, K Soundararajan [3] conjectured that the following two statements are equivalent: lim infβ / log γ = 0; lim infγ + γ logγ = 0, where γ + is the least ordinate of a zero of ζ s with γ + >γ Y Zhang [6] has shown that implies as follows Theorem A Assume RH Let α and α be positive constants satisfying α < π and α >α α π If ρ = / + iγ is a zero of ζ s such that γ is sufficiently large and γ + γ<α log γ, then there exists a zero ρ of s such that ρ ρ <α log γ In this paper, we consider the converse of Theorem A Namely, is it true that implies? M Z Garaev and C Y Yildirim [6] proved a weaker form of the converse; if we assume RH and lim infβ / log γ log log γ = 0, then we have lim inf n γ n+ γ n logγ n = 0 In this article, we prove the following Theorem Assume RH Then the following are equivalent: lim infβ / log γ 0; For any c > 0ands = σ + it with 0 < σ / < c/ log t t > t 0 c, we have ζ s = + Olog t, s ρ where ρ = / + iγ is the zero of ζ closest to s and to the origin, if there are two such

4 4 H Ki Corollary Assume RH and lim infβ / log γ 0 Then for any c > 0ands = σ + it t > t c, we have ζ s = Olog t σ uniformly for / + c/ log t σ σ < Based on our theorems and Soundararajan s conjecture, we speculate as follows Conjecture Assume RH Then the following are equivalent: i For any c > 0ands = σ + it with t > t c, we have ζ s = Olog t σ uniformly for / + c/ log t σ σ < ; ii The negation of ii, ie lim infγ + γ logγ 0 It is known from [, p 99] and [5, Theorem 96A] that for t 0 and σ, one has ζ s = γ t + Olog t, s ρ where the sum is over the ordinates of the complex zeros ρ = β + iγ of ζ s Under RH, one gets ζ s = γ t / + Olog t s ρ Cf [5, p ] Formula and its proof suggest that formula in Theorem is very unrealistic Concerning Corollary, it is worth noting that, under RH, ζ s = Olog t σ 3

5 The Zeros of the Derivative of the Riemann Zeta Function 5 holds uniformly for / + c/ σ σ <, where t 0, c > 0and O depends upon c and σ For the proof, we apply the fact [5, 445] log t Re z log ζ z = O 4 on s z =c/ to the formula s ζ s = πi c s z = loglogt log ζ z s z dz We immediately get equation 3 Note that the same method does not work when / + c/ log t σ σ <, because on s z =c/ log t we lack equation 4 It seems very likely that on Re s = / + c/ log t, s Olog t ζ On the other hand, Corollary follows from a Phragmén Lindelöf argument, anytime one does know that s = Olog t ζ on Re s = / + c/ log t t 0 We find the behavior of the logarithmic derivative of the Riemann zeta function near Re s = / subtle, and so we need a deep observation about the Riemann zeta function near the critical line to establish lim infβ / log γ = 0 In fact, we will see from Theorem 4 in Section that the behavior of s/ζ s onres = / + c/ log t is very much related to 0< γ γ < γ γ, where / + iγ,/ + i γ are complex zeros of ζ s and the sum is over γ

6 6 H Ki From Corollary, we can demonstrate the following Corollary Assume RH and lim infβ / log γ 0 Then for any c > 0ands = σ + it t > t c, we have log t σ log ζ s = O uniformly for / + c/ log t σ σ < Assuming RH, it is known from [5, p ] and [5, Theorem 44B] that for any c > 0ands = σ + it t 0, we have that log t σ log ζ s = O 5 holds uniformly for / + c/ σ σ <, and there exists an absolute constant c > 0 depending on c such that c log t log σ < log ζ s < c log t 6 holds for / <σ / + c/ ; log t arg ζ s = O 7 holds uniformly for / σ / + c/ However, as in the proof of equation 3, one cannot relax the condition / + c/ ofequation5as/ + c/ log t t On the other hand, we have -theorems related to log ζ s near the critical line For these, we refer to Montgomery s results [] and[5, p 09] In particular, Montgomery showed that assuming RH, for / σ<and any real θ, there is a t with T /6 t T such that Re e iθ log ζ s 0 log T σ log log T σ

7 The Zeros of the Derivative of the Riemann Zeta Function 7 See [, p 5] Recently, from random matrix theory, in the case that θ = 0 in the above -result, it is conjectured in [3] that we have the following: max t [0,T] ζ + it = exp + o log T log log T Concerning negative values of log ζ s, we observe that by equations 6 and 7, it is possible that we have log ζ + log t + it >ψt log t for some arbitrarily large values of t, where ψt as t A sharp -result like this implies lim infβ / log γ = 0 We note that such an estimate will follow if we show that equation 5 does not hold uniformly for / + / log t σ / + / We apply our theorem to mean values of the logarithmic derivative of the Riemann zeta function T σ + it ζ dt 0 for σ = / + a/ log T One gets the following from a result of A Selberg [, equation ] Theorem B Assume RH Then T σ + it ζ 0 dt 4a T log T holds, where σ = / + a/ log T and a, a = olog T Going further, we follow Montgomery [0], and let F α, T = T π log T 0<γ, γ T T iαγ γ wγ γ, where β + iγ and β + i γ are zeros of ζ s andwu = 4 4+u Montgomery conjectured that for any fixed A >, F α, T uniformly for α A MH

8 8 H Ki Under RH, D A Goldston, S M Gonek, and H L Montgomery [7] considered T 0 σ + it ζ dt for σ = / + a/ log T as a 0 and proved the following, provided that MH is valid Theorem C Assume RH and MH Then T 0 σ + it ζ dt a T log T holds, where σ = / + a/ log T and a = at 0 sufficiently slowly as T Corresponding to Theorem C, we have the following theorem Theorem Assume RH and lim infβ / log γ 0 Then T 0 σ + it ζ dt = a T log T logπe log T + O a log + O a T holds, where σ = / + a/ log T and a 0 Here O does not depend upon a and T MH implies Montgomery s pair correlation conjecture; that is, 0<γ, γ T 0<γ γ πβ/ log T T β [ π log T 0 sin πu πu ] du Accordingly, MH automatically yields lim infγ + γ logγ = 0 Assuming RH, Theorem A says that lim infβ / log γ 0 implies lim infγ + γ logγ 0

9 The Zeros of the Derivative of the Riemann Zeta Function 9 Thus the assumptions of Theorem C and Theorem are contradictory to each other However, we have the similar conclusion in Theorem C and Theorem Further, a theorem of D A Goldston, S M Gonek, and H L Montgomery [7, Theorem 3] says the following Theorem D Assume RH Then for σ = / + a/ log T and any fixed a > 0, T 0 σ + it ζ dt e a T log T 4a holds, as T if and only if the pair correlation conjecture is true Combining Theorem B and Theorem, we immediately have the following Theorem 3 Assume RH and lim infβ / log γ 0 Then for σ = / + a/ log T, T 0 σ + it ζ dt e a T log T 4a holds, where a 0ora and a = olog T Theorem D and Theorem 3 are, of course, similar However, the conclusion of Theorem D says a much stronger statement than that of Theorem 3 This is perhaps not too surprising since, by Theorem A, we know that the pair correlation conjecture implies lim infβ / log γ = 0 Proofs of Theorem and Corollaries and First, we state a basic fact Lemma Let T > 0 Then we have that the number of zeros of ζ s in0< Im s T is <γ n T = T π log T + Olog T; πe the number of zeros of ζ s int Im s T + isolog T

10 0 H Ki For Lemma, see [, p 98] and [5, Theorem 94] Lemma immediately follows from equation We start with the following theorem Theorem 4 Assume RH and lim infγ n+ γ n logγ n > 0 Then the following three statements are equivalent A lim infβ / log γ > 0 B Let c > 0ands = σ + it For sufficiently large n, wehave ζ s = s ρ n + Olog t, C where γ n +γ n < t γ n++γ n and σ / < c/ log γ n lim sup M n / log γ n <, where M n = 0< γ m γ n γ n γ m Proof of Theorem 4 We may assume that all but finitely many nontrivial zeros of ζ s are simple and on Re s = /, because we assume RH and lim infγ n+ γ n log γ n > 0 C B We recall that ζ s = Olog t + γ t s ρ for Re s andt Lemma Let δ>0 Suppose that lim infγ n+ γ n logγ n >δthenwehave γ n γ m = O log γ n δ m n for sufficiently large n

11 Proof of Lemma We write m n γ m γ n = The Zeros of the Derivative of the Riemann Zeta Function γ m γ n > γ m γ n + 0< γ m γ n γ m γ n 8 = I + II 9 By Lemma, the number of the γ n between t and t + isolog t fort > Then for some C > 0, we get I = C k= γ n +k<γ m γ n +k+ k= C logγ n + k k + k= log γ n + log k k γ m γ n + γ n k> + C k= γ n k γ m <γ n k γ m γ n 0 C logγ n k k k= log γ n k = Olog γ n By the assumption of Lemma, there exists a positive integer n such that γ m+ γ m Using this, for sufficiently large n, we have γ n+k γ n δ log γ m for all m n k δ logγ n + k δ logγ n for γ n+k γ n and k By Lemma, there exists a > 0 such that k a log γ n Thus we get II kδ k a log γ n log γ n < k= = O kδ log γ n log γ n δ We apply this inequality and equation to equation and obtain γ m γ n = O log γ n δ m n for sufficiently large n This proves Lemma

12 H Ki Lemma 3 Choose c > 0 such that lim infγ n+ γ n logγ n > c Define M n t by M n t = 0< γ m γ n t γ m for γ n + c / log γ n t γ n+ c / log γ n Thenwehave M n t = Olog t, where the implied constant is absolute Proof of Lemma 3 Since M n t = 0< γ m γ n < t γ m < 0, M n t is decreasing Thus it suffices to consider the endpoints for the proof By Lemma, assumption C in Theorem 4, and the fact [5, Theorem 9] that the gaps between ordinates of successive zeros tend to 0, we have M n γ n+ c = log γ n γ 0< γ m γ n n+ γ m c log γ n = γ 0< γ m γ n+ n+ γ m c + Olog γ n log γ n = γ 0< γ m γ n+ n+ γ m c + log γ n γ n+ γ m M n+ + Olog γ n = O + Olog γ log γ n γ 0< γ m γ n+ < n+ γ m n+ = Olog γ n Similarly, we have M n γ n + c = Olog γ n log γ n Lemma 3 follows

13 The Zeros of the Derivative of the Riemann Zeta Function 3 Using equation, Lemma, Lemma 3, and C, for ρ n = / + iγ n and s = σ + it t 0, we get ζ s = s ρ n + = s ρ n + = s ρ n + ρ ρn 0< γ t 0< γ m t 0< γ m γ n + Olog t s ρ s ρ m + γm γn > 0< γm t + O s ρ m s ρ m + Olog t 0< γ t + Olog t = + s ρ n s ρ 0< γ m γ n m it γ m + M nt + Olog t i = + O σ + Olog t s ρ n γ m n m γ n in σ / < c/ log γ n By this and Lemma, B follows B A We need the following lemma for this Lemma 4 Assume RH and lim infγ n+ γ n logγ n > 0 Let δ be such that 0 < δ <δ where δ is as in Lemma Suppose β + iγ = 0 for sufficiently large γ If γ γ n δ/ log γ for all n, thenwehave log γ + O = O β log γ δ Proof of Lemma 4 Using formula 8 in [, p 80], we have Re s ζ s = log π Re s s Ɣ Re + Ɣ s + + Re s ρ, 3 where ρ runs through all complex zeros of ζ s We note that any complex zero ρ of ζ s is either / + iγ n or / iγ n for some n, provided that RH is true We assumed that for any n, γ n γ > ρ δ logγ 4

14 4 H Ki By equation 3 we obtain that for s = β + iγ, we get Re ρ s ρ = log π + Re s + s Ɣ Re + Ɣ s + 5 Applying the standard fact [, p73] Ɣ s Ɣs = log s + O, args π >θ>0 s to equation 5, we obtain Re s ρ = log t + O 6 ρ We have Re s ρ = β ρ n= β + γ γ n + β β + γ + γ n β = n= β + γ γ n + O β Thus, by this and equation 6, we obtain log γ β + O = O 7 γ γ n= n Using equation 4 and Lemma, we have n= β γ γ n = O β We insert this to equation 7 and then we get log γ + O = O β log γ δ log γ δ This proves Lemma 4

15 The Zeros of the Derivative of the Riemann Zeta Function 5 Suppose lim infβ / log γ = 0 Then this and Lemma 4 imply that we have sequences ɛ k, ρ nk,and ρ k such that ρ nk = / + iγ nk, ρ k = 0, ρ k ρ n k < ɛ k log γ nk and ɛ k 0 ɛ k > 0 8 Using B we get Thus we obtain that for some c > 0, ρ k ρ n k + Olog γ nk = 0 ρ k ρ n k > c log γ nk Note that c does not depend on ɛ k s By this and equation 8, we obtain ɛ k log γ nk > c log γ nk But this is a contradiction, since ɛ k 0andc > 0 is a fixed real number Thus A follows A C Suppose We rewrite equation as ζ s = 0< γ m γ n lim sup M n log γ n = s ρ m + s ρ n + Olog γ n where we will take s ρ n ɛ/log γ n,withɛ an arbitrarily small positive real number For such s, it follows that s ρ n ζ s s ρ n 0< γ m γ n s ρ m = + O s ρ n log γ n = + Oɛ

16 6 H Ki Since we are assuming that lim sup M n log γ n =,wehave + Oɛ < ɛ log γ n M n +u log γ n, with any real constant u, for infinitely many n Hence for such n and for s ρ n ɛ/log γ n,weobtain s ρ n ζ s s ρ n 0< γ m γ n s ρ m < ɛ M n +ulog γ n log γ n As in the proof of C B, using Lemma when s ρ n ɛ/log γ n,wehave 0< γ m γ n M n s ρ m i ɛ δ log γ n So, with an appropriate constant u the above estimate gives u ɛ, we obtain δ s ρ n ζ s s ρ n s ρ m < s ρ n s ρ m 0< γ m γ n 0< γ m γ n on s ρ n =ɛ/log γ n for infinitely many n Wenotethat s ρ n 0< γ m γ n s ρ m has a zero at s = ρ n Thus Rouché s theorem implies that for some s in s ρ n <ɛ/log γ n, s ρ n ζ s = 0, that is, s = 0 Therefore lim infβ / log γ = 0 Hence A implies C We have completed the proof of Theorem 4 Now we prove Theorem and Corollaries and Proof of Theorem Assume that lim infβ / log γ 0

17 The Zeros of the Derivative of the Riemann Zeta Function 7 Then RH implies that lim infβ / log γ is positive Then by Theorem A, we obtain lim infγ n+ γ n logγ n > 0 Thus by Theorem 4, we have Assume that equation is true If there exist multiple zeros for ζ s, we immediately get a contradiction from equation Thus all zeros of ζ s are simple Suppose that lim infγ n+ γ n logγ n = 0 Then there exists a sequence of natural numbers n k with n k < n k+ such that γ nk + γ nk logγ nk 0, as k Using equation, we have ζ s = + Olog t s ρ nk + = + Olog t s ρ nk at s = / + iγ nk + + γ nk / By this, we obtain or = O log γ nk s ρ nk + s ρ nk γ nk + γ nk = O log γ nk Namely, we have γnk + γ nk log γnk = O This is a contradiction, for lim k γ nk + γ nk logγ nk = 0 Hence by Theorem 4, we have Thus Theorem follows

18 8 H Ki have Proof of Corollary Assume equation We fix c > 0 Then by Theorem 4B, we s = Olog t +3 9 ζ on σ = / + c/log t +3 Using this, it is not hard to see that Corollary follows from a Phragmén Lindelöf argument We give the detailed proof for convenience For the following version of the Phragmén Lindelöf Theorem, we refer to [8, pp ] Phragmén Lindelöf Theorem Let G be an unbounded simply connected region and let f be an analytic function on G Suppose that ϕ is analytic and bounded on G, and never zero Assume that G = A B and that a for every a A, lim sup s a fs M; b for every b B { }and η>0, lim sup s b fs ϕs η M Then fs M for all s in G G = We define sets G and G by { s C : σ } + c log t +3 and G ={s C :0 σ + c, t } We define G by G = G C G We note that G We define fs by fs = s ζ slogs for s G Then fs is analytic on the region G We choose ϕs = exp s Clearly, the function ϕ : G C is an analytic function which never vanishes and is bounded on G By equation 9, there exists M > 0 such that we have fs M 0 on the boundary G

19 The Zeros of the Derivative of the Riemann Zeta Function 9 Claim Assume RH We have ζ s = Olog t +3 s = σ + it G Proof of Claim We may suppose σ<andt 0 Using Lemma, equation, and the fact that σ / c/ log t +3 for s G, we get ζ s = t γ < s ρ + Olog t = O log t t γ < + Olog t = O log t Thus the claim follows Let η>0 Then by the claim, we conclude that in G we have lim sup fs ϕs η = O s lim sup log r exp r η r cos π 4 = 0 With this and equation 0, we see that the functions f and ϕ fulfill the conditions of Phragmén Lindelöf theorem Hence fs M for s G Namely, we obtain s = Olog s ζ for s G In particular, we have s = Olog t ζ uniformly for / + c/ log t σ / + c/ t 0 From this, equation 3, and the fact that for / + c/ log t σ / + c/, log t = O log t σ, we complete the proof of Corollary

20 0 H Ki Proof of Corollary Using equation 5 and Corollary, we have σ σ + it log ζ s = log ζ σ + it σ log t σ σ = O log t σ = O log t σ = O + O + O ζ σ + it d σ σ log t σ d σ log t σ for / + c/ log t σ σ Thus we have proved Corollary 3 Proof of Theorem We let T > andσ = / + a/ log T for a small a We set a n = γ n + γ n Theorem 4 implies that under the assumptions for Theorem, we obtain that for s = σ + it and n =, 3,, ζ s = s ρ n + Olog t 3 for 0 σ / < / log γ n and a n < t a n+ Wewrite T σ + it ζ dt = an+ σ + it n n a n ζ = I + II + O, T dt + σ + it a n ζ dt + O + where a n + < T < a n + Using equation 3, it is easy to see that II = T a n + + Olog T log T σ / + it γ dt = O n + a + olog T

21 We insert equation 3 into I and then we get I = n n = <γ n <T an+ a n O log T The Zeros of the Derivative of the Riemann Zeta Function + Olog T σ / + it γ n dt 3 γ σ / tan n+ γ n σ / + 33 <γ n <T log + γ n+ γ n + O T log T 34 σ / Since lim infγ n+ γ n logγ n > 0, there is α>0 such that γ n+ γ n σ / α a is large as a 0 Using this, Lemma, and the fact that tan x = π + O/x as x, we get <γ n <T tan γ n+ γ n σ / = T 4 log T + OaT log T + Olog T 35 πe We recall that there exists A > 0 such that { # n :0<γ n T, γ n+ γ n λ } = O T log Te Aλ log λ 4, log T uniformly for λ For this we refer to [5]and[5, p 46] Using this formula and Lemma, we can see that <γ n <T log + γ n+ γ n = O T log T σ / m= = O T log T log a log + m e Am log m 4 a We insert this and equation 33 into equation 3 and then we obtain I = a T log T logπe a T log T + O log a T log T + O log T a

22 H Ki Using I and II, we get T σ + it ζ dt = a T log T logπe log T + O a log + O a T Hence Theorem follows Acknowledgment I truly thank Professor C Y Yildirim for his many valuable comments and suggestions on this paper Also, I would like to thank the referee and the editor for carefully reviewing my manuscript and for providing useful suggestions for improving this paper This work was supported by the Korea Science and Engineering Foundation grant funded by the Korea government no R References [] Berndt, B C The number of zeros for ζ k s Journal of the London Mathematical Society, no 970: [] Davenport, H Multiplicative Number Theory, 3rd ed, revised by H L Montgomery Graduate Texts in Mathematics 74 New York: Springer, 000 [3] Farmer, D W, S M Gonek, and C P Hughes The maximal size of L-functions Journal für die Reine und Angewandte Mathematik : 5 36 [4] Feng, S A note on the zeros of the derivative of the Riemann zeta function near the critical line Acta Arithmetica 0, no 005: [5] Fujii, A On the distribution of the zeros of the Riemann zeta function in short intervals Bulletin of the American Mathematical Society 8 975: 39 4 [6] Garaev, M Z, and C Y Yildirim On small distances between ordinates of zeros of ζ s and s International Mathematics Research Notices 007, no 007: 4 [7] Goldston, D A, S M Gonek, and H L Montgomery Mean values of the logarithmic derivative of the Riemann zeta-function with applications to primes in short intervals Journal für die Reine und Angewandte Mathematik : 05 6 [8] Hille, E Analytic Function Theory, vol Boston, MA: Ginn and Co, 96 [9] Levinson, N, and H L Montgomery Zeros of the derivatives of the Riemann zeta-function Acta Mathematica : [0] Montgomery, H L The pair correlation of zeros of the zeta function In Analytic Number Theory, 8 93 Proceedings of Symposia in Pure Mathematics 4 Providence, RI: American Mathematical Society, 973 [] Montgomery, H L Extreme values of the Riemann zeta-function Commentarii Mathematici Helvetici 5 977: 5 8

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