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

Marco Aymone,Jing Zhao,Winston Heap

doi:10.1112/jlms.12421

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.

Highlights

In this paper we investigate a model for the Riemann zeta function provided by a sum of random multiplicative functions
Let (f (p))p be a set of independent random variables uniformly distributed on the unit circle (Steinhaus variables) where p runs over the set of primes and let f (n) = pvp||n f (p)vp
The lower bound (3) suggests that it remains log-normal in this range, which would certainly be in analogy with the zeta function

Summary

Introduction

In this paper we investigate a model for the Riemann zeta function provided by a sum of random multiplicative functions. The lower bound (3) suggests that it remains log-normal in this range, which would certainly be in analogy with the zeta function. Since the zeta function at height T oscillates on a scale of roughly 1/ log T (which can be seen either by considering its zeros or its approximation by a Dirichlet polynomial) one might expect that by sampling it at T log T independent points on the interval [T, 2T ] one can pick up the maximum From this point of view (4) represents a model for maxt∈[T,2T ]. The lower bound of Theorem 4 better displays the multiplicative nature of the problem It suggests the sum is potentially being dictated by its Euler product since (1 − f (p)p−1/2)−1 ≈ exp( f (p)p−1/2).

Proof of Corollary 1

Moment bounds

Tail bounds

Almost sure bounds

Omega bounds