The zeros of the derivative of the Riemann zeta function near the critical line

Transcription

1 arxiv:math/07076v [mathnt] 5 Jan 007 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 haseoyonseiackr February, 008 Abstract 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 non-real zero ρ, equality being possible only at a multiple zero of ζs) In this paper we prove that lim inf β /) log γ 0 if and only if for any c > 0 and s = σ + it with σ / < c/log t t 0) ζ s) = + Olog t), s ρ where ρ = / + iγ is the closest zero of ζs) to s and the origin We also show that if lim inf β /) log γ 0, then for any c > 0 and s = σ + it t 0), we have log t) σ ) log ζs) = O log log t uniformly for / + c/log t σ σ < Introduction The Riemann hypothesis RH) states that the real part of any nonreal zero of the Riemann zeta function ζs) is / A Speiser [3] has a theorem which says that RH is equivalent to the nonexistence of nonreal zeros of s) in Re s) < /

2 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 [8] 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 ordinate less than T is 0<γ T = T π log T + Olog T) 4πe Under RH, K Soundararajan [] 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 log T #{ρ : β / + ν/ log T, 0 < γ T } > 0 π In his remarkable work, Y Zhang [5] showed that not only unconditionally there exists a ν > 0 such that a positive proportion of zeros of s) are in the region σ / < ν/ log T, but also assuming RH and a strong hypothesis of the distribution of zeros of ζs), ν can be arbitrary small Recently, Feng [5] proved the Zhang s second theorem only assuming the strong hypothesis Thus it is very probable that lim inf β /)log γ = 0 Assuming the truth of RH, K Soundararajan [] conjectured that the following two statements are equivalent: i) lim infβ /) logγ = 0; ii) lim infγ + γ) log γ = 0 where γ + is the least ordinate of a zero of ζs) with γ + > γ Y Zhang [5] has shown that ii) implies i) 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 ζs) 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 i) implies ii)? Concerning this problem, we have the following

3 3 Theorem Assume RH Then the following are equivalent: ) lim infβ /) logγ 0; ) For any c > 0 and s = σ + it with σ / < c/ log t t 0) ζ s) = + Olog t), s ρ where ρ = / + iγ is the closest zero of ζs) to s and the origin; Corollary Assume RH and lim infβ /) log γ 0 Then, for any c > 0 and s = σ+it t 0), we have ζ s) = O log 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 > 0 and s = σ + it t 0), we have uniformly for / + c/ log t σ σ < ; ζ s) = O log t) σ) ii) The negation of ii), ie, lim infγ + γ) log γ 0 We briefly introduce why ) in Theorem doesn t seem possible We let s = σ + it for real numbers, σ, t It is known in [3, p 99] and [4, Theorem 96A)] that for t 0 and σ we have ζ s) = γ t + Olog t), ) s ρ where the sum is over the ordinates of the complex zeros ρ = β + iγ of ζs) However, assuming RH, we may expect a stronger result Under RH we have ζ s) = γ t /log log t + Olog t) ) s ρ for t 0 For this we refer to [4, p )] According to the formula ), we can see that the formula ) in Theorem is very unrealistic, because σ / < c/ log t in ) doesn t seem probable instead of γ t / loglog t in )

4 4 Concerning Corollary, it is worth noting that under RH ζ s) = O log t) σ) 3) holds uniformly for / + c/ log log t σ σ <, where t 0, c > 0 and O depends upon c and σ For the proof of it, we apply the fact [4, 445)] to the formula s) ζs) = log ζz) πi c s z) dz s z = log log t Then we immediately get 3) In proving 3), one cannot relax the condition / + c/ log log t as + c t ) log t With this information, it is very likely that on Re s) = / + c/ log t t 0), s) Olog t) ζ On the other hand, Corollary follows from a Phragmén-Lindelöf argument, provided that we have s) = Olog t) ζ on Re s) = / + c/ log t t 0) Thus 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)/ζs) on Re s) = / + c/ log t is very much related to 0< γ γ < γ γ, where / + iγ, / + i γ are complex zeros of ζs) and the sum is over γ From Corollary, we can demonstrate the following Corollary Assume RH and lim infβ /) logγ 0 Then, For any c > 0 and s = σ+it t 0), we have ) log t) σ log ζs) = O log log t uniformly for / + c/ log t σ σ <

5 5 Assuming RH, it is known in [4, p )] and [4, Theorem 44B)] that for any c > 0 and s = σ + it t 0), we have ) log t) σ log ζs) = O 4) log log t holds uniformly for / + c/ log log t σ σ <, and there exists an absolute constant c > 0 depending on c such that ) c log t log log t log ) < log ζs) < c log t 5) σ log log t log log t holds for / < σ / + c/ log log t; arg ζs) = O ) log t log log t 6) holds uniformly for / σ / + c/ log log t However, as in the proof of 3), one cannot relax the condition / + c/ log log t of 4) as / + c/ log t t ) On the other hand, we have Ω-theorems related to log ζs) near the critical line For these, we refer to Montgomery s results [0] and [4, 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 ζs) ) 0 log T) σ log log T) σ See [0, p 5] Recently, from random matrix theory, in the case that θ = 0 in the above Ω-result, it is conjectured in [4] that we have the following: ) ) max t [0,T] ζ + it = exp + o)) log T log log T Concerning negative values of log ζs), we observe that by 5) and 6), it is possible that we have log ζ + ) log t + it log t > ψt) log log t for some arbitrarily large values of t, where ψt) as t A sharp Ω-result like this implies lim infβ /) logγ = 0 Namely, we note that it will follow if we show that 4) does not hold uniformly for / + / log t σ / + / log log t Thus, Corollary is useful in investigating the horizontal behavior of zeros of the derivative of the Riemann zeta function We apply our theorem to mean values of the logarithmic derivative of the Riemann zeta function: σ + it) ζ dt for σ = / + a/ log T 0 We may get the following from a result of A Selberg [, equation )]

6 6 Theorem B Assume RH Then 0 σ + it) ζ dt 4a T log T holds where σ = / + a/ log T and a, a = olog T) We introduce more studies on mean values of the logarithmic derivative of ζs) Following Montgomery [9], let Fα, T) = T log T π 0<γ, γ T T iαγ γ) wγ γ), where β + iγ and β + i γ are zeros of ζs) and wu) = 4 4+u Montgomery conjectured that for any fixed A >, MH) Fα, T) uniformly for α A Under RH, D A Goldston, S M Gonek and H L Montgomery [7] considered σ + it) ζ dt 0 for σ = / + a/ log T as a 0 and proved the following, provided that MH) is valid Theorem C Assume RH and MH) Then σ + it) ζ 0 dt a T log T holds where σ = / + a/ log T and a = at) 0 sufficiently slowly) as T We have the following as in Theorem Theorem Assume RH and lim infβ /) log γ 0 Then σ + it) ζ dt = a T log T logπe) + O a log ) )) + O log T a T 0 holds where σ = / + a/ log T and a 0 Here O doesn t depend upon a and T MH) implies Montgomery s pair correlation conjecture That is, T β ) sin πu π log T du πu 0<γ, γ T 0<γ γ πβ/log T 0

7 7 Clearly MH) implies Assuming RH, Theorem A says that lim infγ + γ) log γ = 0 lim infβ /) logγ 0 implies lim infγ + γ) log γ 0 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, 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, 0 σ + it) ζ dt e a T log T 4a holds where a 0 or a and a = olog T) Apparently, Theorem D and Theorem 3 are similar However the conclusion of Theorem D says a much stronger statement than that of Theorem 3 We notice that the pair correlation conjecture implies lim infβ /) logγ = 0 Thus the conclusion of Theorem 3 under lim infβ /) logγ 0 is instructive to understand the behavior of the mean value of the logarithmic derivative of the Riemann zeta function Proof of Theorem and Corollaries, We arrange the zeros of the Riemann zeta function ζs) on the upper half-plane as ρ, ρ, with ρ n = β n + iγ n and 0 < γ γ, where it appears precisely m times consecutively in the above sequence, if a zero is multiple with multiplicity m RH is that β n = / for any n =,, 3, We state basic facts

8 8 Proposition Let T > 0 Then, we have: ) The number of zeros of ζs) in 0 < Im s) T is = T π log T πe + OlogT); <γ n T ) The number of zeros of ζs) in T Im s) T + is Olog T) For Proposition ), see [3, p 98] and [4, Theorem 94] Proposition ) immediately follows from ) We start with the following theorem Theorem 4 Assume RH and lim infγ n+ γ n ) log γ n > 0 Then the following three statements are equivalent: A) lim infβ /) logγ > 0; B) Let c > 0 and s = σ + it For sufficiently large n, we have ζ s) = + Olog t), s ρ n where γ n +γ n < t γ n++γ n and σ / < c/ log γ n ; C) 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 ζs) 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 for Re s) and t s) = Olog t) + ζ s ρ γ t Proposition Let δ > 0 Suppose that lim infγ n+ γ n ) log γ n > δ Then we have log ) γ n γ m ) = O γ n δ for sufficiently large n m n

9 9 Proof of Proposition We write γ m γ n ) = m n γ m γ n > =I + II k= γ n+k<γ m γ n+k+ γ m γ n ) + 0< γ m γ n γ m γ n ) ) By Proposition ), the number of γ n s between t and t + is Olog t) for t > Then, for some C > 0, we get I = γ m γ n ) + γ m γ n ) C k= k= C logγ n + k) k log γ n + log k k + γ n k> + C k= k= γ n k γ m<γ n k C logγ n k) k log γ n k = Olog γ n ) By the assumption of Proposition, 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 log γ n ) for γ n+k γ n and k Here there exists a a > 0 such that k a log γ n Thus we get log ) γ n II kδ k alog γ n log γ n ) < k= ) = O kδ log γ n We apply this inequality and ) to ) and then we obtain log ) γ m γ n ) = O γ n δ m n for sufficiently large n This proves Proposition Proposition 3 Choose c > 0 such that lim infγ n+ γ n ) log γ n > c Define M n t) by M n t) = t γ m 0< γ m γ n for γ n + c / log γ n t γ n+ c / log γ n Then we have where the implied constant is absolute M n t) = Ologt), δ

10 0 Proof of Proposition 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 Proposition and our assumption C), we have M n γ n+ c ) = log γ n γ n+ γ m c log γ n Similarly, we have Proposition 3 follows = = 0< γ m γ n 0< γ m γ n+ 0< γ m γ n+ M n+ + Olog γ n ) =O log γ n =Olog γ n ) γ n+ γ m c + Olog γ n ) log γ n γ n+ γ m c + log γ n γ n+ γ m 0< γ m γ n+ < M n γ n + c ) = Olog γ n ) log γ n + Ologγ γ n+ γ m ) n+ ) Using Proposition ), Proposition 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 + = s ρ n + = s ρ n + O ρ ρ n 0< γ t 0< γ m γ n 0< γ m γ n 0< γ m γ n m n + Olog t) s ρ s ρ m + γ m γ n > 0< γ m t + O s ρ m s ρ m σ γ m γ n ) 0< γ t s ρ m + Ologt) + Olog t) it γ m ) + M nt) + Olog t) i ) + Olog t)

11 in σ / < c/ log γ n By this and Proposition, B) follows B) A) We need the following proposition for this Proposition 4 Assume RH and lim infγ n+ γ n ) log γ n > 0 Let δ be such that 0 < δ < δ where δ is as in Proposition Suppose β + iγ ) = 0 for sufficiently large γ If γ γ n δ/ log γ ) for all n, then we have Proof of Proposition 4 We set log γ + O) = O β ) log ) γ δ ξs) = ss ) π s/ Γs/)ζs) for any s C It is known that for some constants A and B, ξs) = e A+Bs s ) e s ρ, ρ ρ where ρ runs through all zeros of ξs) See [3, p 80] and [4, p 30] for this We note that any zero ρ of ξs) is either / + iγ n or / iγ n for some n, provided that RH is true By the product formula of ξs), we get Then we have s) ζs) = log π s Γ s + ) Γ s + ) + B + s ρ + ρ ρ Re s) ζs) = log π Re s We assumed that for any n, γ n γ > By 3) we obtain that for s = β + iγ, we get ρ Re Γ Re s ρ = log π + Re s + Applying the standard fact [3, p 73] Γ s) Γs) s + ) Γ s + ) + Re s ρ 3) ρ δ log γ 4) Re Γ = log s + O ), args) π > θ > 0) s s + ) Γ 5) s + )

12 to 5), we obtain We have Re s ρ = log t + O) 6) ρ 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 6), we obtain log γ + O) = O Using 4) and Proposition, we have n= β γ γ n ) = O We insert this to 7) and then we get This proves Proposition 4 log γ + O) = O n= β γ γ n ) ) β ) ) log γ ) δ β ) ) log γ ) δ 7) Suppose lim infβ /) logγ = 0 Then this and Proposition 4 implies that we have sequences ǫ k, ρ nk and ρ k such that ρ nk = / + iγ nk, ρ k ) = 0, ρ k ρ n k < Using B), we get Thus we obtain that for some c > 0, ρ k ρ n k + Ologγ nk ) = 0 ρ k ρ n k > c log γ nk Note that c doesn t depend on ǫ k s By this and 8), we obtain ǫ k log γ nk > c log γ nk ǫ k log γ nk and ǫ k 0 ǫ k > 0) 8)

13 3 But this is a contradiction, since ǫ k 0 and c > 0 is a fixed real number Thus A) follows A) C) Suppose We write lim sup M n log γ n = ζ s) = Olog γ n) + s ρ n + 0< γ m γ n s ρ m for s ρ n ǫ/ log γ n Let ǫ be an arbitrarily small positive real As in the proof of C) B), using Proposition, we obtain 0< γ m γ n M n s ρ m i = Olog γ n ) on s ρ n ǫ/ log γ n Then we can see that on s ρ n = ǫ/ log γ n we have s ρ n) ζ s) s ρ n) s ρ m 0< γ m γ n < ǫ M n + Olog γ n )) log γ n = s ρ n) s ρ m for infinitely many n s We note that s ρ n ) 0< γ m γ n s ρ m ) 0< γ m γ n has a zero at s = ρ n Thus Rouché s theorem implies that for some s in s ρ n < ǫ/ log γ n, s ρ n ) ζ s ) = 0, ie, 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, Proof of Theorem Assume that lim infβ /) logγ 0 Then RH implies that lim infβ /) logγ is positive Then, by Theorem A, we obtain lim infγ n+ γ n ) log γ n > 0 Thus, by Theorem 4, we have ) )

14 4 Assume ) is true If there exist multiple zeros for ζs), we immediately get a contradiction from ) Thus all zeros of ζs) are simple Suppose that lim infγ n+ γ n ) log γ n = 0 Then there exists a sequence of natural numbers n k with n k < n k+ such that as k Using ), we have γ nk + γ nk ) log γ nk 0 ζ s) = + Ologt) s ρ nk + = + Olog t) s ρ nk at s = / + iγ nk + + γ nk )/ By this, we obtain or Namely, we have = Ologγ nk ) s ρ nk + s ρ nk γ nk + γ nk = Ologγ nk ) γ nk + γ nk ) log γ nk = O) This is a contradiction, for lim k γ nk + γ nk ) log γ nk = 0 Hence, by Theorem 4, we have ) ) Thus Theorem follows Proof of Corollary Assume ) We fix c > 0 Then, by Theorem 4 B), we have s) = O log 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 [, p 38] Phragmén-Lindelöf Theorem Let G be a simply connected region and let f be an analytic function on G Suppose there is an analytic function ϕ : G C which never vanishes and is bounded on G If M is a constant and G = A B such that a) for every a in A, lim sup fs) M; s a b) for every b in B, and η > 0, lim sup fs) ϕs) η M; s b then fs) M for all s in G

15 5 Here G = G = the boundary of G if G is bounded, G = G { } if G is unbounded and the limit superior of fs) as s a, is defined by lim sup s a fs) = lim sup{ fs) : s G Ba, r)}, r 0 + where Ba, r) = {s C : s a < r} If a =, Ba, r) is the ball in the metric of C = C { } We apply this theorem for proving ) 3) We define sets G and G by G = {s C : σ + We define G by We note that G We define fs) by c log t + 3) } and G = {s C : 0 σ + c, t } G = G C G ) fs) = s) ζs) logs 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 9), there exists M > 0 such that we have fs) M 0) on the boundary G Claim We have for s = σ + it G ζ s) = O log t + 3) ) Proof of Claim We may suppose σ < and t 0 Using Proposition ), ) and the fact that σ / > c/ log t + 3) for s G, we get Thus, Claim follows ζ s) = t γ < =O log t =O log t ) + Olog t) s ρ t γ < + Olog t)

16 6 Let η > 0 Then, by 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 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 for s G In particular, we have s) = Olog s ) ζ s) = Olog t) ζ uniformly for / + c/ log t σ / + c/ log log t t 0) From this, 3) and the fact that for / + c/ log t σ / + c/ log log t, we prove Corollary log t) σ = Olog t), Proof of Corollary Using 4) and Corollary, we have σ σ + it) log ζs) = log ζσ + it) σ ζ σ + it) d σ log t) σ ) σ ) =O + O log t) eσ d σ log log t σ log t) σ ) ) log t) σ =O + O log log t log log t ) log t) σ =O log log t for / + c/ log t σ σ Thus, we prove Corollary 3 Proof of Theorem We let T > and σ = / + a/ log T for a small a We set a n = γ n + γ n Theorem 4 implies that under the assumption of Theorem, we obtain that for s = σ +it and n =, 3,, ζ s) = + Olog t) 3) s ρ n

17 7 for 0 σ / < / logγ n and a n < t a n+ We write σ + it) ζ dt = an+ σ + it) ζ dt + n n a n =I + II + O), a n + σ + it) ζ dt + O) where a n + < T < a n + Using 3), it is easy to see that II = + Olog T) log ) T σ /) + it γ n +) dt = O + olog T) a a n + We insert 3) into I and then we get I = an+ + Olog T) n n a n σ /) + it γ n ) dt = γ σ / tan n+ γ n σ /) + <γ n<t O log T log + γ ) ) n+ γ n + O T log T ) σ /) <γ n<t Since lim infγ n+ γ n ) log γ n > 0, there is a β > 0 such that γ n+ γ n σ /) β a 3) 33) is large as a 0 We recall Proposition ) = T π log T + Olog T) 34) πe <γ n<t Using 33), 34) and the fact that tan x = π + O/x) as x, we get <γ n<t tan γ n+ γ n σ /) = 4 T log T T logπe) + OaT log T) + Olog T) 35) 4 We recall that there exists a A > 0 such that <γ n<t #{n : 0 < γ n T, γ n+ γ n λ log T } = O T log Te Aλ log λ) 4 ), 36) uniformly for λ For this we refer to [6] and [4, p 46] Using 34) and 36), we can see that log + γ ) ) n+ γ n = O T log T log + m ) e Am log m) 4 σ /) a m= = O T log T log a ) 37)

18 8 We insert 35) and 37) into 3) and then we obtain I = a T log T logπe) T log T + O log ) a a T log T) + Olog T/a) Using I and II, we can get σ + it) ζ dt = a T log T logπe) + O a log ) )) + O log T a T Hence Theorem follows 4 Acknowledgment I truly thank Professor C Y Yildirim for his many valuable comments and suggestions on this paper References [] B C Berndt, The number of zeros for ζ k) s), J London Math Soc) 970), [] J B Conway, Functions of one complex variable, nd ed, Graduate Texts in Mathematics, Springer-Verlag, New York, 978 [3] H Davenport, Multiplicative Number Theory, 3rd ed, revised by HL Montgomery, Graduate Texts in Mathematics 74, Springer-Verlag, New York, 000 [4] D W Farmer, S M Gonek and C P Hughes, The maximal size of L-functions, J Reine Angew Math, to appear [5] S Feng, A note on the zeros of the derivative of the Riemann zeta function near the critical line, Acta Arith 0 005), [6] A Fujii, On the distribution of the zeros of the Riemann zeta function in short intervals, Bull Amer Math Soc 8 975), 39 4 [7] D A Goldston, 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, J Reine Angew Math ), 05 6 [8] N Levinson and H L Montgomery, Zeros of the derivatives of the Riemann zetafunction, Acta Math ), 49 65

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