MAXIMA OF A RANDOMIZED RIEMANN ZETA FUNCTION, AND BRANCHING RANDOM WALKS

• 成果类型: Article
• 署名作者: Arguin, Louis-Pierre; Belius, David; Harper, Adam J.
• 署名单位: Universite de Montreal; City University of New York (CUNY) System; Baruch College (CUNY); City University of New York (CUNY) System; New York University
• 刊物名称: ANNALS OF APPLIED PROBABILITY
• ISSN/ISSBN: 1050-5164
• DOI: 10.1214/16-AAP1201
• 发表日期: 2017
• 页码: 178-215
• 关键词: convergence; LAW

摘要

A recent conjecture of Fyodorov-Hiary-Keating states that the maximum of the absolute value of the Riemann zeta function on a typical bounded interval of the critical line is exp{loglogT - 3/4logloglogT + O(1)}, for an interval at (large) height T. In this paper, we verify the first two terms in the exponential for a model of the zeta function, which is essentially a randomized Euler product. The critical element of the proof is the identification of an approximate tree structure, present also in the actual zeta function, which allows us to relate the maximum to that of a branching random walk.