Maxima of a randomized Riemann zeta function, and branching random walks

By L. Arguin, David Belius, Adam J Harper. 2015

84 citations recorded by the source corpus.

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 $\exp\{\log \log T -\frac{3}{4}\log \log \log T+O(1)\}$, 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.

Subfields: Algebra and Number Theory, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/68b17dcb59bde3180389430d227140b0799809a0

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In Algebra and Number Theory

In Mathematical Physics

Citations in the corpus

Extrema of log-correlated random variables: Principles and Examples (2016, methodology): “xactly of the order 1=n. This is the content of the ballot theorem that we now state precisely. The reader is referred to [1,2,24] for more details. The result quoted here is the version appearing in [7]. Theorem 5. Let (Sn)n0 be a Gaussian random walk with increments of mean 0 and variance ˙2 >0”
The Riemann zeta function and Gaussian multiplicative chaos: Statistics on the critical line (2016, methodology): “[40] A. Harper: Sharp conditional bounds for the moments of the Riemann zeta function. arXiv:1305.4618 [41] D. R. Heath-Brown: Fractional moments of the Riemann zeta-function.”
Maximum of the Characteristic Polynomial of Random Unitary Matrices (2015, methodology): “andom walk, or hierarchical eld, in the Fourier decomposition of the characteristic polynomial. This is inspired by a branching structure in the Euler product of the Riemann Zeta function employed in [4]. In Section 1.2, it is explained how branching random walk heuristics provide an alternative jus”
On the extreme values of the Riemann zeta function on random intervals of the critical line (2016, methodology): “This proposition is very similar to results given by Arguin, Belius and Harper in [5]: see for example Proposition 2.”
Extreme value statistics of correlated random variables: A pedagogical review (2019, background): “This, in turn, has generated a recent interest in mathematics [247–249], in connection with random matrices and branching processes.”
MOMENTS OF RANDOM MULTIPLICATIVE FUNCTIONS, I: LOW MOMENTS, BETTER THAN SQUAREROOT CANCELLATION, AND CRITICAL MULTIPLICATIVE CHAOS (2017, background): “on at nearby t), but on the other hand one is interested in proving sharper results about multiplicative chaos (e.g. bounds that are sharp up to constants). See the paper of Arguin, Belius and Harper [1] for results on the maximum of a random model for logζ(1/2+it), which again is close to log|F(1/2”