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

By Joseph Najnudel. 2016

81 citations recorded by the source corpus.

In the present paper, we show that under the Riemann hypothesis, and for fixed $$h, \epsilon > 0$$h,ϵ>0, the supremum of the real and the imaginary parts of $$\log \zeta (1/2 + it)$$logζ(1/2+it) for $$t \in [UT -h, UT + h]$$t∈[UT-h,UT+h] are in the interval $$[(1-\epsilon ) \log \log T, (1+ \epsilon ) \log \log T]$$[(1-ϵ)loglogT,(1+ϵ)loglogT] 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 $$\mathfrak {R}\log \zeta (1/2 + it)$$Rlogζ(1/2+it) is at most $$\log \log T + g(T)$$loglogT+g(T) with probability tending to 1, g being any function tending to infinity at infinity.

Subfields: Algebra and Number Theory, Discrete Mathematics and Combinatorics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/792faea0e6f69b18ec6a7bec4b0bee8323869cb8

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In Algebra and Number Theory

In Discrete Mathematics and Combinatorics

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Citations in the corpus

Maximum of the Riemann Zeta Function on a Short Interval of the Critical Line (2016, methodology): “The identification of an approximate branching random walk structure within the sum ∑ p≤X p −s was used in [3] to study the extrema of a random model of the zeta function, and in the subsequent works regarding the large values of characteristic polynomials [2,12,26,33] and of the zeta function [32].”
The Riemann zeta function and Gaussian multiplicative chaos: Statistics on the critical line (2016, background): “[3, 68] (see also [2, 70, 22] for related results in the setting of random matrices).”