Partial sums of random multiplicative functions and extreme values of a model for the Riemann zeta function

TL;DR

This paper models the Riemann zeta function using partial sums of a random multiplicative function, analyzing its tail behavior, moments, and maximum, providing insights consistent with conjectures about the zeta function's extreme values.

Contribution

It introduces a probabilistic model for the zeta function's behavior, analyzing its tail distribution, moments, and maximum, aligning with existing conjectures.

Findings

Determined the tail distribution and high moments of the model.

Estimated the maximum of multiple independent sums, matching conjectured zeta maxima.

Provided bounds indicating significant cancellation beyond square root level.

Abstract

We consider partial sums of a weighted Steinhaus random multiplicative function and view this as a model for the Riemann zeta function. We give a description of the tails and high moments of this object. Using these we determine the likely maximum of $T \log T$ independently sampled copies of our sum and find that this is in agreement with a conjecture of Farmer--Gonek--Hughes on the maximum of the Riemann zeta function. We also consider the question of almost sure bounds. We determine upper bounds on the level of squareroot cancellation and lower bounds which suggest a degree of cancellation much greater than this which we speculate is in accordance with the influence of the Euler product.