MOMENTS OF THE RIEMANN ZETA FUNCTION ON SHORT INTERVALS OF THE CRITICAL LINE

• 成果类型: Article
• 署名作者: Arguin, Louis-Pierre; Ouimet, Frederic; Radziwill, Maksym
• 署名单位: City University of New York (CUNY) System; Baruch College (CUNY); City University of New York (CUNY) System; California Institute of Technology
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/21-AOP1524
• 发表日期: 2021
• 页码: 3106-3141
• 关键词: GAUSSIAN MULTIPLICATIVE CHAOS; random energy models; maximum; bounds; sums

摘要

We show that as T -> infinity, for all t is an element of [T, 2T] outside of a set of measure o(T), integral(log theta T)(-log theta T)vertical bar zeta(1/2 + it +ih)vertical bar(beta) dh = (log T)(f)((beta)+)(o(1))(theta), for some explicit exponent f(theta)(beta), where theta > -1 and beta > 0. This proves an extended version of a conjecture of Fyodorov and Keating (Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 372 (2014) 20120503, 32). In particular, it shows that, for all theta > -1, the moments exhibit a phase transition at a critical exponent beta(c)(theta), below which f(theta)(beta) is quadratic and above which f(theta)(beta) is linear. The form of the exponent f(theta) also differs between mesoscopic intervals (-1 < theta < 0) and macroscopic intervals (theta > 0), a phenomenon that stems from an approximate tree structure for the correlations of zeta. We also prove that, for all t is an element of [T, 2T] outside a set of measure o(T), max(vertical bar)h vertical bar <= log(theta) T vertical bar zeta(1/2 + it + ih)vertical bar = (log T) (m(theta)+o(1)), for some explicit m(theta). This generalizes earlier results of Najnudel (Probab. Theory Related Fields 172 (2018) 387-452) and Arguin et al. (Comm. Pure Appl. Math. 72 (2019) 500-535) for theta = 0. The proofs are unconditional, except for the upper bounds when theta > 3, where the Riemann hypothesis is assumed.