On the extreme values of the Riemann zeta function on random intervals of the critical line

• 成果类型: Article
• 署名作者: Najnudel, Joseph
• 署名单位: University System of Ohio; University of Cincinnati
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s00440-017-0812-y
• 发表日期: 2018
• 页码: 387-452
• 关键词: minimal position; CONVERGENCE; maximum; LAW

摘要

In the present paper, we show that under the Riemann hypothesis, and for fixed h, epsilon > 0, the supremum of the real and the imaginary parts of log.(1/ 2 + i t) for t. [ UT -h, UT + h] are in the interval [(1 -epsilon) log log T, (1 + epsilon) log log T] with probability tending to 1 when T goes to infinity, U being a uniform random variable in [ 0, 1]. This proves a weak version of a conjecture by Fyodorov, Hiary and Keating, which has recently been intensively studied in the setting of random matrices. We also unconditionally show that the supremum of epsilon log.(1/ 2 + i t) is at most log log T + g(T) with probability tending to 1, g being any function tending to infinity at infinity.