Sharp conditional bounds for moments of the Riemann zeta function

Abstract

We prove, assuming the Riemann Hypothesis, thatfor any fixed k ≥ 0 and all large T . This is sharp up to the value of the implicit constant.Our proof builds on well known work of Soundararajan, who showed, assuming the Riemann Hypothesis, thatT for any fixed k ≥ 0 and ǫ > 0. Whereas Soundararajan bounded log |ζ(1/2 + it)| by a single Dirichlet polynomial, and investigated how often it attains large values, we bound log |ζ(1/2+it)| by a sum of many Dirichlet polynomials and investigate the joint behaviour of all of them. We also work directly with moments throughout, rather than passing through estimates for large values.

Sharp lower bounds for moments of $ζ'(ρ)$

“…We shall instead apply the lower bounds principal of W. Heap and K. Soundararajan [9] in the proof of Theorem 1.1. The proof also uses the arguments by A. J. Harper in [8] and by S. Kirila in [12]. Combining Theorem 1.1 and the above mentioned result of S. Kirila in [12], we immediately obtain the following result concerning the order of magnitude of J k (T ).…”

“…We follow the ideas of A. J. Harper in [8] and the notations of S. Kirila in [12] to define for a large number M depending on k only,…”

“…The approach taken by M. B. Milinovich and N. Ng to obtain their result concerning (1.2) follows from a simple and powerful method developed by Z. Rudnick and K. Soundararajan in [22,23] for establishing lower bounds for moments of families of L-functions, while the approach taken by S. Kirila follows from a method of K. Soundararajan [24] together with its refinement by A. J. Harper [8] for establishing upper bounds for moments of families of L-functions.…”

Maxima of log-correlated fields: some recent developments

“…is an integer for β ∈ N. Additionally, Ramachandra [129] and Heath-Brown [91] have established a lower bound M T (β) ≫ (log T 2π ) β 2 for positive, rational β. Radziwiłł and Soundararajan [128] proved lower bounds of the correct order for all real β ≥ 1. Upper bounds of the correct form M T (β) ≪ (log T 2π ) β 2 are known, conditionally on the Riemann hypothesis, due to arguments of Soundararajan [136] and Harper [88].…”

“…Theorem 3.8 then follows from proposition 3.9 by setting L = K = β and α 1 = • • • = α 2β = 1 (using the rotational invariance of the Haar measure). Thus Using (88), the Schur polynomial s λ (1 n ) is equal to the number of semistandard Young tableaux of shape λ with entries in {1, . .…”