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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.