MAXIMUM OF THE RIEMANN ZETA FUNCTION ON A SHORT INTERVAL OF THE CRITICAL LINE

Transcription

1 MAXIMUM OF HE RIEMANN ZEA FUNCION ON A SHOR INERVAL OF HE CRIICAL LINE L-P ARGUIN, D BELIUS, P BOURGADE, M RADZIWI L L, AND K SOUNDARARAJAN Abstract We prove the leading order of a conecture by Fyodorov, Hiary and Keating, about the maximum of the Riemann zeta function on random intervals along the critical line More precisely, if t is uniformly distributed in,, then max ζ + iu = + o log log t u log with probability converging to as Introduction Maximum of the Riemann ζ function on large and short intervals Bounding the maximum of the Riemann zeta function on the critical line has been the source of many investigations since Lindelöf, who conectured that for any ε > 0, we have ζ + i = O ε as Among the many arithmetic consequences of the Lindelöf hypothesis we highlight the existence of primes in all intervals x, x+x /+ε for all x large enough, and in almost all intervals of the form x, x + x ε he current best bound towards the Lindelöf hypothesis states that ζ + it + t 3/84+ε, see 7 Conditionally on the Riemann hypothesis, Littlewood 4 showed that ζ + i = O exp C log, log log for some constant C > 0 Apart from the value of the constant C 3, 36, 0, this bound remains the best that is known here has been more progress on lower bounds for the maximal size of the zeta-function he first result is due to itchmarsh, who proved that for any α <, and large enough, max ζ/ + is s 0, explogα L-P A is supported by NSF grant DMS-5344, and a Eugene M Lang Junior Faculty Research Fellowship D B is grateful for the hospitality of the Courant Institute during visits when part of this work was carried out P B is supported by NSF grant DMS M R is grateful to NSERC for partially funding this research KS is partly supported by a grant from the NSF, and a Simons Investigator grant from the Simons Foundation

2 L-P ARGUIN, D BELIUS, P BOURGADE, M RADZIWI L L, AND K SOUNDARARAJAN his result was improved to log max ζ/ + is exp c s 0, log log in 6 under the Riemann hypothesis, then the constant c was improved in 4 and, unconditionally, in 35 he best lower bounds on the maximum were recently obtained in 6: for any c < /, max s 0, ζ + is log log log log exp c log log Previous results with differing methods were established in 6, 4, 35 From and, the actual size order of this maximum is unclear, and its asymptotics are not known to follow from conectures in number theory However, interesting probabilistic arguments 3 suggest that max log ζ/ + is s 0, log log log here are however dissenting views, advocating that is closer to the maximal size of the Riemann zeta-function see end of 3 hese arguments are motivated by Waldspurger s theorem and an analogy between integral weight and half-integral weight modular forms Although the global maximum up to height can hardly be tested numerically, the maximum along random short intervals was very precisely conectured by Fyodorov, Hiary and Keating, and their prediction is supported by numerics 5, 6 his conecture states that if t is random, uniform in 0,, then 3 max ζ + is = log log 3 4 log log log + X, t s log where the random variable X converges weakly as, to an explicit distribution Our main result is a proof for the first order asymptotics in 3 heorem Let t be uniform in, For any ε > 0, as we have { } meas t : ε log log < max log t s ζ + is < + ε log log While completing this work, we learned that the above result as well as the analogue for Im log ζ was independently proved in 8 under the assumption of the Riemann hypothesis Extrema of log-correlated fields Fyodorov, Hiary and Keating s conecture was motivated by a connection with random matrices his analogy has been a source of many investigations, for example at the microscopic level since Montgomery s pair correlation conecture 5, and the Keating Snaith conecture about the moments of the Riemann zeta function he prediction 3 relies on the analogy at a different, mesoscopic scale Indeed, Selberg proved that log ζ is normally distributed on the critical axis, with variance For convenience we state the conecture 3 with a random interval of length, without loss of generality: only the parameters of the limiting distribution of X are supposed to depend on this fixed length

3 MAXIMUM OF HE RIEMANN ZEA FUNCION ON A SHOR INERVAL OF HE CRIICAL LINE3 of order log log 34 His central limit theorem has been extended to evaluation points at close distance in 8: for example, if t is uniform on, and 0 < h < depends on t, the covariance between log ζ + it and log ζ + it + h is of order 4 h, log max, log where log is a natural threshold corresponding to the typical spacing between ζ zeros A parallel story holds for the logarithm of the characteristic polynomial of N N Haardistributed unitary matrices, log P N z : for z =, it is asymptotically Gaussian with variance log N, and for two points on the unit circle within distance z z = h, the covariance between log P N z and log P N z is of order log max h, N, analogously to 4, see 8 Fyodorov, Hiary and Keating gave a very precise conecture for the maximum of {log P N z, z = } by relying on the replica method, and techniques from statistical mechanics predicting extreme values in disordered systems 4, 7, 8 Finally, assuming the logarithmic covariance form characterizes the extrema, they proposed the asymptotics 3 he above Fyodorov-Hiary-Keating analogy, about extreme value theory, has recently been proved in a variety of cases For a probabilistic model of the Riemann zeta function the leading order of the maximum on short intervals was obtained in 0, and the second order in 3 For the characteristic polynomial of random unitary matrices, the asymptotics of the maximum at first order and then second order 9 are known, together with tightness of the third order in the more general context of circular beta ensembles In the context of Hermitian invariant ensembles, the first order of the maximum of the characteristic polynomial was proved in 3 and precise conectures can be found in 9 heorem and its conditional analogue in 8 are the first results about the maxima of ζ itself, with the only source of randomness being the choice of the interval Moreover, in connection with the prediction from 5, 6 that log ζ behaves like a real log-correlated random field, we note that 33 recently proved that ζ converges to a complex Gaussian multiplicative chaos Finally, this work builds on, and adds to, the efforts to develop extreme value theory of correlated systems Such statistics are expected to lie on the same universality class for any covariance of type 4 his class includes the two-dimensional Gaussian free field, branching random walks, cover times of random walks, Gaussian multiplicative chaos, random matrices and the Riemann zeta function We do not give here a list of the many rigorous works on this topic in recent years, pointing instead to and the references therein 3 About the proof he short proof of the upper bound in heorem is given in section We only rely on a Sobolev inequality and classical second moment estimates for ζ and ζ he proved upper bound is actually stronger than as stated in heorem : for any function f diverging to +, as, we have meas { max t s } log ζ + is < log log + f his result is obtained in a different way in 8, also unconditionally he lower bound in heorem is proved in section 3 As a first step, we show that we only need to prove the lower bound for a sufficiently long but finite length Dirichlet sum,

4 4 L-P ARGUIN, D BELIUS, P BOURGADE, M RADZIWI L L, AND K SOUNDARARAJAN which corresponds to the first explog ε terms in the expansion of log ζ over primes, for some ε > 0 his step makes use of ideas developed to give an alternative proof of Selberg s central limit theorem in 3 Specifically we need to develop a maximal analogue of the argument in 3 he first observation is that it suffices to prove the lower bound for the local maxima of ζσ + it with σ slightly off the half-line, as large values off the halfline propagate to the half-line he second point is the construction of a special mollifier Ms which is shown to have two properties: On the one hand for almost all t we show that ζσ + iumσ + iu for all t u On the other hand we show that for almost all t we have Mσ + iu p X p σ+iu for all t u and some suitably chosen cut-off X he combination of these two properties implies that for almost all t we have ζσ + iu p X p σ+iu for all t u hus the problem is reduced to understanding the local maximum of p X p σ+iu, which is a short sum over the primes A key idea in the proof of the lower bound for the local maximum of p X p σ+iu is the identification of an approximate branching random walk in this Dirichlet polynomial his idea was used in 3 to study the extrema of a random model of the zeta function, and in the subsequent works regarding the extremes of characteristic polynomials,, 3, 9 and the zeta function 8 he conceptual picture is explained in detail in 3, Once the branching structure appears, we follow a second moment method which goes back to Bramson s study of branching Brownian motion 9; specifically we use Kistler s robust multiscale refinement from, in a manner close to his method requires large deviation estimates and speed of convergence to the normal distribution for our Dirichlet sums, when evaluated at distinct points For that purpose, the Fourier and Laplace transforms with high arguments are evaluated through an expansion over moments Large moments are typically hard to approximate, a problem we overcome here following a decomposition from 30: over a good subset B of, the first log log moments provide an accurate approximation of the characteristic function, and the complement of B has small enough measure It should be possible to obtain more refined results by pushing the parameter K to infinity, but we have not attempted to carry this out For the rest of the paper, we will think of t has a uniform random variable on, We will write accordingly P for meas and E for We will also write f = Og if f f is bounded and f = og if 0 Finally, we will sometimes write for short g g f g when f = Og Proof of the upper bound he upper bound is a direct consequence of unconditional moment bounds and of the Sobolev-type inequality in Lemma A Proposition If V = V tends to infinity as, then P max ζ/ + iu > V log = O/V = o t u Proof Chebyshev s inequality implies 5 P ζ/ + iu > V log max t u V log E max ζ/ + iu t u

5 MAXIMUM OF HE RIEMANN ZEA FUNCION ON A SHOR INERVAL OF HE CRIICAL LINE5 Lemma A bounds the second moment of the maximum by E max ζ/ + iu E ζ/ + iu / E ζ / + iu / +E ζ/ + iu t u he following bounds of the second moment of ζ and its derivative are known unconditionally see, eg, 37 and : E ζ/ + it log We conclude that he proposition follows from this and 5 E ζ / + it log 3 E max ζ/ + iu log t u 3 Proof of the lower bound he lower bound of heorem is proved in two main steps First, it is shown that the maximum on a short interval of log ζ is close to the maximum of a Dirichlet polynomial slightly off the critical axis his is the content of Proposition 3 Second, a lower bound for the maximum of Dirichlet polynomials on an interval is proved using the robust approach of in Proposition 3 he following notation will be used throughout the section Following, we will fix a large integer K = Kε For this K, we take 6 σ 0 = + log 3 K he primes will be divided into ranges as follows log 7 J = explog K, explog + K, =,, K, and J 0 =, explog K We also denote 8 X = explog K Finally, the relevant Dirichlet polynomials are 9 P u = Re p J p σ 0+iu, = 0,, K he lower bound in our theorem follows from the two main propositions stated below Proposition 3 For any ε > 0 and K = Kε large enough, P max log ζ/ + iu > ε log log P t u max t u 4 = P u > ε log log Proposition 3 For any K > 3 and 0 < λ <, 0 P u : t u, P u > λk log log for all K 3 = + o + o

6 6 L-P ARGUIN, D BELIUS, P BOURGADE, M RADZIWI L L, AND K SOUNDARARAJAN Proof of heorem Since on the event of Proposition 3, we have = P u > λ 3 log log for some u with t u K herefore it suffices to take K large and λ close to in terms of ε to get the lower bound of the theorem using Proposition 3 3 Proof of Proposition 3 he proof of the proposition is divided into three lemmas In the first, we bound the maximum on the critical axis by the ones off axis Lemma 33 Let ε > 0, V > and σ + log / ε hen P max ζ / + it > V P max ζ σ + it > V + o t u t u 4 Proof Recall that, for any t R, we can write ζσ + it as an average on the critical line using the Poisson kernel: ζσ + it = ζ / + iu σ / π u t + σ / du We denote the above Poisson kernel centered by f t u Consider t, such that max v ζ σ + it + v > V 4 We write v for the v, that achieves the maximum for this t Equation implies 4 4 V < ζσ + it + v In particular, for this t, it must be that or t+/ t / ζ / + iu + v f t udu ζ / + iu + v f t udu > V, t,t+ c ζ / + iu + v f t udu > V Since v and f 4 tu is a probability density centered at t, the first case implies that max t u ζ / + it > V herefore it remains to prove that the set {t : ζ / + iu + v f t udu > V } t,t+ c has small probability Again, since v 4, it suffices to show that the probability of t 4,t+ 4 c ζ / + iu f t udu > V is small

7 MAXIMUM OF HE RIEMANN ZEA FUNCION ON A SHOR INERVAL OF HE CRIICAL LINE7 By Chebyshev s inequality, the probability is smaller than σ / ζ / + iu E π V t >/4 u t + σ / du log ε E ζ / + it + v dv, 4, v 4 c where second inequality follows by definition of σ and Jensen s inequality applied on 4, dv 4 c 8 v he last expression is Olog ε o see this, note that E ζ / + it + v is Olog for v /, say For v > /, one can use the fact that ζ/ + it = Ot /4 see, eg heorem 5 in 37 to conclude that the integral over this range of v is For the second lemma, we take the following approximation for the inverse of ζ: Ms = n µnan n s, where for n square-free µn = ωn and ωn is the number of distinct prime factors, and for n non-square-free µn = 0 he factor an equals if all primes factors of n are smaller than X and Ωn 00K log log =: ν, where Ωn is the number of primes factors of n he following shows that the maxima of ζ and M are close Lemma 34 For any ε > 0, we have P max Mσ 0 + iuζσ 0 + iu > ε = + o t u Proof We closely follow 3, Section 4, adapting it to different scalings Assume we can prove that, uniformly in σ > + log + K +ε, 3 Mσ + isζσ + is ds = o hen Lemma 34 follows by Chebyshev s inequality and Lemma A o prove 3, we first consider the cross term in the square expansion We integrate along the rectangular contour with vertices σ + i, σ + i, + i, + i, and use the estimates ζ = O /4 37 and M = O ε on the horizontal parts his gives ζσ + ismσ + isds = where we used the simple expansion We therefore proved 4 ζ + ism + isds = + Mσ + isζσ + is ds = ζ + ism + isds + O /4+ε = + O /4+ε, m,n:mn amµm mn mn is ds = + O Mσ + isζσ + is ds + O /4+ε,

8 8 L-P ARGUIN, D BELIUS, P BOURGADE, M RADZIWI L L, AND K SOUNDARARAJAN and we now turn to the evaluation of the above second moment: Mσ + isζσ + is ds = h,k µhµkahak hk σ is h ζ σ + is ds k It can be estimated based on the following result see 3, Lemma 4: for any h, k and / < σ, we have 5 is h ζ σ + is ds = ζσ k h, k σ + hk he contribution of the above error terms is bounded by t π σ h, k σ ζ σ ds hk + O σ+ε minh, k σ+ε h,k minh, k ahak hk σ ν σ+ε ah σ+ε = O σ+ε h p X he contribution of the first term in 5 is 6 ζσ h,k ahakµhµk hk σ h, k σ If we omit the constraint Ωh, Ωk ν in the above sum, we obtain for A a subset of P X, the set of primes smaller than X, we write π A = p A p: h,k:p hk p X µhµk hk σ h, kσ = A + B πσ π A π B σ A B A,B P X = A P X,C A,D P X A A + C + D = π A π D σ p σ p X where we decomposed B = C D and used C A C = 0 unless A = he error consisting in terms such that Ωh > ν or Ωk > ν is bounded using 57 by h,k:p hk p X Ωh>ν = e ν A P X,D P X A µhµk h, k σ hk σ e ν h,k:p hk p X e A e ν π A π D σ e A π σ A P X A µhµk h, k σ hk σ = e ν p P X e Ωh + e = O p σ log 98

9 MAXIMUM OF HE RIEMANN ZEA FUNCION ON A SHOR INERVAL OF HE CRIICAL LINE9 where we used 53 We therefore proved that 6 = ζσ = p X + O p σ log 98 p>x + O p σ σ log 98 = + O, log 96 where we used 5 to estimate log p>x p σ = p>x p σ + O/X his is negligible whenever σ log X here σ X = log /K Finally, the contribution from the second term in 5 is shown to be negligible as in 3 For the last reduction, define the Dirichlet polynomials 7 P s = n J Λn n s log n, = 0,, K hen again, the maxima of M and exp P are close Lemma 35 We have K P max t u Mσ 0 + iu exp P σ 0 + iu > log = + o =0 Proof We defined the truncated exponential 8 Ms = k ν k k! P s k Consider the good set { 9 U = t : max P σ 0 + iu 0 } log log, 0 K 3 t u K/ he choice of upper bound is motivated by the logarithmic correlations 4 hese imply that, for each, the values of P are almost perfectly correlated for points at a distance less than log + K, and close to independent for points farther than log K apart herefore, at a heuristic level, one expects that the maximum of P corresponds to the one of log + K independent Gaussian variables of variance log log, which is of the order of K + log log Here we pick a bound that holds simultaneously for all Note that on the K set U, M is arbitrarily close to exp P since k k! k>ν P s k k>ν k! log log k log 00

10 0 L-P ARGUIN, D BELIUS, P BOURGADE, M RADZIWI L L, AND K SOUNDARARAJAN Moreover, the complement has small probability, since by Chebyshev s inequality applied with l = log log and the Sobolev s inequality A with L = log, we get PU c 0 log log E max P σ l 0 + iu l K / t u L 0 sup E P log log l σ + iu l + o K / σ σ 0 L he bound on the moments in Lemma B yields E P σ + iu l l! K log log l for all uniformly for σ σ 0 L We conclude that PU c log l l e l l / log log l K 0 log log log l 4 K / With the above observations, the proof of the lemma is reduced to showing that P max Mσ 0 + iu Mσ 0 + iu > log 3 = + o t u Again, this follows by Chebyshev s inequality and the Sobolev s inequality A if E Mσ + iu Mσ + iu 0 uniformly for σ σ 0 L o see this, note that M can be written as a Dirichlet sum n bnn s where bn, by expanding the powers of P Moreover bn = anµn whenever Ωn ν, and bn = 0 if n X ν his is because the difference between M and M only comes from the prime powers in P hus if we write cn = bn µnan, the L difference becomes m,n cncm mn σ Em/n it If m n, since logm/n mn /, the sum is smaller than X ν / by an upper bound on the number of terms If m = n, then we have that the sum is n r ν + r n p + r log 99r p p X > Ωn>ν p\n p X where we used Rankin s trick 57 with < r < his concludes the proof of the lemma Proof of Proposition 3 Note that the difference between the Dirichlet polynomials P and P is given by the prime powers Qσ 0 + iu := K =0 P u p X p = σ 0+iu p + O, σ 0+iu p X

11 MAXIMUM OF HE RIEMANN ZEA FUNCION ON A SHOR INERVAL OF HE CRIICAL LINE since the higher powers are summable by 5 herefore by Chebyshev s inequality and the Sobolev inequality A we have P max Qu > log log log t u 4 E Qσ0 + it + E Qσ log log log 0 + it / E Qσ 0 + it / A quick calculation with Lemma B shows that the parenthesis is O, so that the probability goes to 0 Furthermore, we have on the set U in 9 that max t u /4 P 0 log log K / for = 0 and K with probability going to By taking K = Kε large enough, these two observations imply P max Re t u 4 K =0 P u ε log log P max t u 4 = P u ε log log he two events appearing in Lemma 34 and 35 have probability + o, and so does their intersection he two lemmas were proved for t u but holds the same way for a smaller interval If u is such that Re K =0 P u 3ε/ log log, we must have on these two events ζσ 0 + iu M σ 0 + iu log + e P + olog ε Altogether, we have so far shown P max ζσ 0 + iu log ε P t u 4 max t u 4 = P u ε log log + o o conclude, it suffices to redefine ε and apply Lemma 33 with σ 0 and V = log ε 3 Proof of Proposition 3 he proof of the proposition is based on large deviation estimates of the variables P u, t u, for one point and two points, see Propositions 39 and 30 hese estimates are derived using large moments of Dirichlet polynomials given in Lemma B, and by estimating the Laplace-Fourier transform of the polynomials using the moments, see Proposition 38 he first step is to show that the moments of sums of P τ are very close to Gaussian moments Proposition 36 Let ξ, and ξ, in C such that ξ, ξ log 6K hen for any n log K, we have for τ, τ, n E {ξ P t + τ + ξ P t + τ } 0 = n!! n {s ξ + ξ + ρ τ, τ ξ ξ } + Oe log K +o

12 L-P ARGUIN, D BELIUS, P BOURGADE, M RADZIWI L L, AND K SOUNDARARAJAN for s = p σ0 and ρ τ, τ = p σ0 cos τ τ log p p J p J he odd moments are Oe log K under the same condition on n Note that without the error term the right-hand side of 0 is precisely what the n-th moment would be if P t+τ, P t+τ were ointly Gaussian with variance s and covariance ρ τ, τ, and uncorrelated for different Using 5 one obtains the bounds s = log log K +Olog K, ρ τ, τ = O log K if τ τ log + m Of course s ρ τ, τ s and in fact ρ τ, τ is close to s if τ τ log m, since then the cosine in the sum is close to Proof of Proposition 36 Let ap = a p = 0 for p / J and ap = ξ p iτ + ξ p iτ p σ 0, a p = ξ p iτ + ξ p iτ p σ 0 for p J so that {ξ P τ + ξ P τ } = {app it + a pp it } p apa p = ξ + ξ + ξ ξ cos τ τ log pp σ 0 if p J In this case, the function J appearing in Lemma B takes the form J apa pz = + apa pz + O apa p z 4 4 p p = exp z p σ0 ξ + ξ + ξ ξ cos τ τ log p F X z, p J for F X z a function which is analytic in a neighborhood of 0, satisfies F X 0 = and whose derivatives are at 0 are uniformly bounded by apa p log 8K p log 8K e log K p J p J he claim 0 follows from Lemma B by taking the n-th derivative note that the exponential term is exactly the moment generating function of a Gaussian and noting that the terms involving a derivative of F X z contribute at most Oe log K Also error term from Lemma B is X n / e log K under the assumption n log K It is necessary to introduce a cutoff to obtain a precise comparison of the Dirichlet polynomials with Gaussians With this in mind, we introduce the set 3 Bτ = { t : P t + τ log 4K K 3}

13 MAXIMUM OF HE RIEMANN ZEA FUNCION ON A SHOR INERVAL OF HE CRIICAL LINE3 Note that if the moments of a random variable Y grow no faster than Gaussian moments with some σ > 0 up to l L, then 4 PY > x exp x σ whenever x L his is proved using Chebyshev s inequality and optimizing over l σ ogether with Proposition 36 this observation yields the following: Lemma 37 Let τ We have 5 PBτ c exp log K log log Proof A union bound gives PBτ c = PP t + τ > log 4K Recall that K is fixed We use the Gaussian bound 4 and Note that log /4K log /K his yields the claimed bound On Bτ, we can derive precise bounds for the Fourier-Laplace transforms of the P Proposition 38 Let ξ C, K 3, such that ξ log 6K hen for τ, 6 E exp ξ P t + τ Bτ = + Oe log 8K exp ξ s = Moreover, if τ τ, then for ξ C, K 3, with ξ log 6K, we also have E exp = 7 ξ P t + τ + ξ P t + τ Bτ Bτ = + Oe log 8K exp = {s ξ + ξ + ρ τ, τ ξ ξ } Proof he one point bound 6 follows from the two-point bound 7 with ξ = 0 he proof of the latter consists in developing the exponential and use the Gaussian moments in Proposition 36 hese hold up at least to moments log K he restriction to the set B takes care of the higher moments he difference between the restricted moments and the ones calculated in the corollary induce some error when computing the moments his error gets worse as the moments grow With this in mind, we choose to cut the development at N = 0 log 3K Write for simplicity P for P t + τ and P for P t + τ, and similarly for Bτ and Bτ he exponential can be written as E exp ξ P + ξ P B B = 8 n N n! E n ξ P + ξ P B B + n>n s = n! E n ξ P + ξ P B B

14 4 L-P ARGUIN, D BELIUS, P BOURGADE, M RADZIWI L L, AND K SOUNDARARAJAN By the definition of B, the bound on ξ and ξ, and the choice of N, the second term is bounded by 9 n>n n! ξ + ξ log 4K n Klog 6K + 4K N N! exp Klog 6K + 4K e log 3K, since Klog 6K + 4K N/0 = log 3K For the first sum in 8, the unrestricted even moments are given by 0 in Proposition 36: 30 n { } E ξ P + ξ P = n!! s ξ + ξ + ρ τ, τ ξ ξ n+oe log K For the odd moments, we simply have by Proposition 36: n 3 E ξ P + ξ P e log K We claim that for any n N, the unrestricted moments are close to the unrestricted ones: 3 E ξ P + ξ P n B B = E Indeed, Cauchy-Schwartz inequality implies ξ P + ξ P n + Oe log 4K E ξ P + ξ P n B B c E ξ P + ξ P n / PB c / From 30, we see that the first term is bounded by / n! log n 4K n N N log 4K N n! Equation 3 then follows from the choice of N and Lemma 37 Using 3 together with 30 and 3 as well as 9, we get 33 E exp = n N/ = exp n>n/ ξ P + ξ P n n! Bτ Bτ { } n s ξ + ξ + ρ τ, τ ξ ξ + Oe log 4K { } s ξ + ξ + ρ τ, τ ξ ξ + Oe log 4K he second inequality comes from the fact that for ξ, ξ log 6K, { } n s n! ξ + ξ + ρ τ, τ ξ ξ e log 4K

15 MAXIMUM OF HE RIEMANN ZEA FUNCION ON A SHOR INERVAL OF HE CRIICAL LINE5 he additive error in 33 can be absorbed in the main term since e log 4K e log 8K { exp s ξ + ξ + ρ τ, τ ξ ξ } he statements of Proposition 38 are used to get precise large deviation estimates on the variables P Define the event 34 τ = {t : P t + τ > x, K 3} We first prove a two point bound for τ and τ that are at mesoscopic distance Proposition 39 Let τ and τ such that log m+/k τ τ < log m/k for some 0 m K 3 We have for x = s m 35 P τ τ exp x + x s s = =m+ Proof Since the right-hand side of 5 is of lower order than the right-hand side of 35, it suffices to prove a bound on P τ τ Bτ Bτ For simplicity, we write P = P t + τ, P = P t + τ, B = Bτ and B = Bτ Note that by definition of the set, the probability P B B is bounded by E exp = β P + P = B B exp = for any β > 0 Using 7 we get that this is at most c exp β s + ρ τ, τ β x for all 0 < β < log 6K For m we have the trivial bound ρ τ, τ s m + we have by that ρ τ, τ = O, so that the probability is at most m c exp 4β s + β s β x + O β = =m+ = = β x =m+ By setting β = x/s for m + and β = x/s for m we obtain 35 and for We also need precise large deviation bounds for one point τ and two points τ and τ that are at near macroscopic distance Proposition 30 For τ we have for x = s 36 P τ + o Moreover, if τ τ log K, then 0 e x+y s dy πs 37 P τ τ + o P τ P τ

16 6 L-P ARGUIN, D BELIUS, P BOURGADE, M RADZIWI L L, AND K SOUNDARARAJAN he proof consists of inverting the Fourier-Laplace transform of Proposition 38 We will do this by directly using the following lemma used in and based on 5, Corollary 5 Lemma 3 Let d Let µ and ν be two probability measures on R d with Fourier transform ˆµ and ˆν here exists constant c > 0 such that for any function f : R d R with Lipschitz constant C and for any R, N > 0, fdµ fdν C R d R N + f { RN d ˆµ ˆν N,N d + µ R, R d c + ν R, R d c } d here will be a small error in the inversion if there is a broad range of parameters where the Fourier transforms are close In the present setting, the errors on the probability need to be olog to compensate the number of points in the discretization he range of the Fourier transform in Proposition 38 is thus a priori insufficient o improve the error, it is necessary to make a change of measure to make the value λ log log typical K Proof of Proposition 30 We start with the one-point bound 36 It suffices to lower bound P τ Bτ We drop the dependence on τ for simplicity Define the measure Q by the Radon-Nikodym derivative dq dp = exp β P Eexp β P B, β = x s he probability of B under P can be written as P B = E e β P B e β x E Q e β P x = + oe x s E Q e β P x by Proposition 38 and the choice of β It remains to prove that 38 E Q e β P x + o 0 e x y s Consider the functions g defined by { 0 if y 0 g y = e β y if y log 64K, e y s πs with a linear interpolation on the interval 0, log 64K By construction, g is Lipschitz with constant log 64K Moreover, E Q e β P x EQ g P x dy he expectation on the right can now be compared to the expectation over independent Gaussians variables with variance s via Fourier inversion he Fourier transform of the

17 MAXIMUM OF HE RIEMANN ZEA FUNCION ON A SHOR INERVAL OF HE CRIICAL LINE7 distribution of P x, K 3 under Q is 39 E Q e i t P x = e i t x E e β +it P B E e β P = +Oe log 8K exp t B s, for t log 3K by Proposition 38 he right side corresponds to the Fourier transform of independent Gaussian variables of variance s he difference between the Fourier transforms on that range is Oe log 8K We apply Lemma 3 with d = K 3, N = R = log 3K and f = g his yields 40 E Q gp x log 64K + log 6K log 3K e log y g 0 y e s dy πs 8K + e log 3K + e log 3K log 64K, where the last two terms in the first inequality come from the measure of R, R d c under the Gaussian and Q measure; they are bounded respectively by a standard Gaussian tail bound and the corresponding bound under Q, which can be obtained from the exponential Chebyshev inequality and the exponential moment bound 39 with t purely complex he claim 38 follows from 40 and the fact that 4 since y s g y e dy πs e x y s e y s πs e x s 0 x /s e x y s e y s πs dy log 64K, e u/ du s log π x 64K his concludes the proof of 38 he proof of the two-point bound 37 is done similarly As in the proof of Proposition 39, it suffices to prove a bound on P τ τ Bτ Bτ For simplicity, we write P for P τ and B for Bτ he right change of measure is now dq dp = he probability P B B becomes 43 E e β P +P B B exp β P + P Eexp β P + P B B, β = x e β x E Q e β P +P x s It follows from 7 that 44 E exp ξ P + ξ P = B B = + Olog 4K exp ξ + ξ s, =

18 8 L-P ARGUIN, D BELIUS, P BOURGADE, M RADZIWI L L, AND K SOUNDARARAJAN for ξ, ξ log 6K, where we have used that ξ ξ ρ τ, τ log 8K log K = Olog 4K, by We thus expect the variables P, P, K 3 to be approximatively independent Gaussians of variance s By 44, the quantity in 43 equals + oe x s E Q e β P +P x We now show that 45 E Q e β P +P x + o 0 e x y s e y s πs dy his is proved by using Lemma 3 with d = K 3, N = R = log 3K where g is now {0 if y log 64K g y = e β y if y > 0, and f = g with linear interpolation in-between his function has Lipschitz constant log 64K, and by definition, 46 E Q e β P +P x E Q g P x g P x he Fourier transform under the Q-measure is 47 E Q e i t P +t P x = e i t +t x E e β +it P +β +it P B B E e β P +P B B = + Olog 4K exp t + t s, for t, t log 6K by 44 Again, the right side corresponds to the Fourier transform of pairs of independent Gaussian variables of variance s he difference between the two Fourier transform is Olog 4K Lemma 3 directly implies E Q g P x g P x g 0 ye y s dy 48 log 64K + log 6K log 3K log 4K + e log 3K + e log 3K log 64K, similarily to in 40 he claim 45 follows from 48 and 46 he error coming from g is controlled as in 4 his concludes the proof of 45 and of the proposition It is now possible to finish the proof of Proposition 3

19 MAXIMUM OF HE RIEMANN ZEA FUNCION ON A SHOR INERVAL OF HE CRIICAL LINE9 Proof Without loss of generality, we suppose that log is an integer Consider a finite set τ l of log points that are equidistant on, As mentioned in, we have that s = log log + o herefore, for λ <, the probability in the statement is greater or K equal to P l τ l Moreover, by a direct application of the Cauchy-Schwarz inequality the Paley-Zygmund inequality, this is bounded below by E l 49 τ l l l E l P τ = τ l l,l P τ l τ l On one hand, by 36 and 4, the sum in the numerator is bounded below by 50 P τ l + o log l log e x s log log / 0 e x +y s dy πs On the other hand, the sum in the denominator can be split into three parts he first is over pairs with τ l τ l log K here are at most log such pairs, therefore by 37, 5 P τ l τ l + o P τ l τ l τ l log K In view of 49 and 50, it remains to show that the sum over pairs such that τ l τ l > log K are negligible with respect to the numerator here are at most pairs log K with log K < τ l τ l log K Proposition 39 with m = 0 implies that the sum over these pairs is log x K e s log K P τ l = o P τ l, = l where the second inequality is from the bound 50 Finally, there are at most log m K pairs with log m+ K < τ l τ l log m K By Proposition 39, the sum of the probabilities over these pairs is m log m K exp x + x x m = log mk x exp exp s s s s =m+ log exp l = l = x s = log m K λ λ his implies that the sum over m from to K 3 is O log K exp x = s which is also negligible with respect to the numerator by 50 his concludes the proof of the proposition

20 0 L-P ARGUIN, D BELIUS, P BOURGADE, M RADZIWI L L, AND K SOUNDARARAJAN Appendix A Sobolev-type inequalities he following Sobolev-type inequalities will be needed several times throughout the paper Lemma A Let f be a differentiable function on 0, such that ft = ot / and f t = ot / hen / max u, ft+u dt ft dt / f t dt + ft dt Proof For any t, and any u,, integration yields the inequality ft + u = t+u t t+ t f s fs ds + t+ t+u f s fs ds + ft + ft + f s fs ds + ft + ft + herefore, by taking the maximum over u and averaging on t, we get E max ft + u E f t + s ft + s ds + u, E ft + E ft + By assumption on f and f, we have any s,, E f t + s E f t and E ft + s E ft he claim follows from this obsrvation and Cauchy-Schwarz inequality applied on the first term Lemma A Let p > 0, let f be an analytic function in the strip σ 0 < σ < σ hen for any L >, and σ with σ 0 + /L σ σ /L, max fσ + t u iu p dt L sup σ σ /L,σ+/L Proof o prove the lemma it suffices to notice that, max fσ + iu max t u t u /L /L L + fσ + iu p and by subharmonicity of f p, the above is /L L fσ + x + iy p dydx L x /L L t y /L /L he claim follows by integrating over t /L fσ + it p dt t+ t fσ + x + iy p dydx Appendix B Elementary Number heory Estimates he Prime Number heorem states that, cf 7, #{p x : p prime} = x dy + log x Oxe c log y

21 MAXIMUM OF HE RIEMANN ZEA FUNCION ON A SHOR INERVAL OF HE CRIICAL LINE In particular, this implies 5 P p Q Q p = dx + log P σ P x σ Oe c log x When σ = / + δ and δ log Q <, it is easy to check that this yields 53 = log log Q log log P + Oδ log Q pσ P p Q his is the case in particular for the choice of σ 0 6 where δ = log + 3 K Q is the upper bound of the set J given in 7 and the largest he next lemma confirms the heuristics that the Dirichlet polynomials are very close to polynomials where p it is replaced by IID uniform variables on the circle his result appears elsewhere, in particular in 8, proof of Proposition 3 We prove it for completeness Lemma B Let X and let ap, p primes and a p, p primes be two sequences in C with ap, a p for all p and ap = a p = 0 for p > X hen for k N we have k E app it + a pp it X = z k J apa pz k + O z=0 p where J z = n 0 zn / n n! In particular, the expression is O X k / for odd k Proof he expectation can be written as k ap lp it l + a p l p it l p,,p k E l= Denote the number of prime factors of n with multiplicities by Ωn We define the multiplicative function gq α = /α! if q is prime If we write n = q α qr αr for the prime decomposition of n, the above sum becomes 54 Ωn=k k!gn E k l= p aq lq it l + αl a q l ql it Let N l be independent Binomial random variable with parameters α l, / Denote the expectation over these by E Note that by expanding the power α l each term can be written as aq lq it l + αl a q l ql it = Eaq l N l a q l α l N l e it log qn l α l l he expectations E and E can be interchanged Moreover, unless N l = α l / for all l we have that i log l qn l α l l c In this case the factor coming from the integration of the X k exponential in t is bounded by X k / hese observations imply altogether the following: E E aq l N l a q l α l N l e it log qn l α l i = aq l a q l α l/ αl X k + O α l α l / l l

22 L-P ARGUIN, D BELIUS, P BOURGADE, M RADZIWI L L, AND K SOUNDARARAJAN where we use the bound on ap, a p and the convention that α α/ = 0 if α/ is not an integer Consider the multiplicative function f defined by fp α = apa p α/ α α α/ With these observations 54 becomes X k 55 k!fngn + O, Ωn=k where for the error term we used the fact that there are at most X k terms in the sum he condition Ωn = k can be expressed using a contour integral 56 k!fngn = k! fngnz Ωn dz πi z k+ Ωn=k Since the summand is a multiplicative function, the finite sum may be expressed as an Euler product fngnz Ωn = + apa pz l l = J apa pz l l! l n p l p he lemma follows by putting this back in 56 and using Cauchy s formula A similar argument can be used to prove an upper bound on the moment, see Lemma 3 in 36 Lemma B Let ap be a sequence of complex coefficients and σ / hen for X and k N such that X k log, we have E ap k ap k k! p σ+it p σ p /k p /k Note that when the sum does not involve only primes but also prime powers like in the case of P i one can apply the Lemma with the choice of ap = αp+βp /p / for example with αp the coefficient at the primes and βp the coefficients at squares of primes he following observation called Rankin s trick will be used often: 57 ann α x α ann α an 0, α > 0 n>x an x α n>x n n= References L-P Arguin, Extrema of log-correlated random variables: principles and examples, Preprint arxiv: L-P Arguin, D Belius, and P Bourgade, Maximum of the characteristic polynomial of random unitary matrices, to appear in Comm Math Phys 05 3 L-P Arguin, D Belius, and AJ Harper, Maxima of a randomized Riemann zeta function, and branching random walks, Preprint arxiv:

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24 4 L-P ARGUIN, D BELIUS, P BOURGADE, M RADZIWI L L, AND K SOUNDARARAJAN 3 M Radziwill and K Soundararaan, Selberg s central limit theorem for log ζ + it, Preprint arxiv: K Ramachandra and A Sankaranarayanan, On some theorems of Littlewood and Selberg, J Number heory , E Saksman and C Webb, he Riemann zeta function and Gaussian multiplicative chaos: statistics on the critical line, preprint, arxiv: A Selberg, Contributions to the theory of the Riemann zeta-function, Arch Math Naturvid , K Soundararaan, Extreme values of zeta and L-functions, Mathematische Annalen , , Moments of the Riemann zeta-function, Annals of Math , EC itchmarsh, he theory of the Riemann zeta-function, Second Edition, Oxford Univ Press, New York 986 Department of Mathematics, Baruch College and Graduate Center, City University of New York, USA address: louis-pierrearguin@baruchcunyedu Institute of Mathematics, University of Zurich, Switzerland address: davidbelius@cantabnet Courant Institute, New York University, USA address: bourgade@cimsnyuedu Department of Mathematics and Statistics, McGill University, Canada address: maksymradziwill@mcgillca Department of Mathematics, Stanford University, USA address: ksound@stanfordedu