Moments of Random Multiplicative Functions and Truncated Characteristic Polynomials

Abstract

Abstract. We give an asymptotic formula for the 2kth moment of a sum of multiplicative Steinhaus variables. This was recently computed independently by Harper, Nikeghbali and Radziwi l l. We also compute the 2kth moment of a truncated characteristic polynomial of a unitary matrix. This provides an asymptotic equivalence with the moments of Steinhaus variables. Similar results for multiplicative Rademacher variables are given.

Moments of Random Multiplicative Functions, I: Low Moments, Better Than Squareroot Cancellation, and Critical Multiplicative Chaos

“…In particular, we find that E| n≤x f (n)| ≍ √ x (log log x) 1/4 , which proves Helson's [15] somewhat surprising conjecture that the first moment should be o( √ x), and disproves the counter-conjecture of Harper, Nikeghbali and Radziwi l l [13] (see also Conjecture 1 of Heap and Lindqvist [14]). Theorem 1 also implies a negative answer to the so-called embedding problem for Dirichlet polynomials (see Question 2 of [24], or Problem 2.1 of [25]) for all exponents 0 < 2q < 2.…”

Shifted Moments of ‐functions and Moments of Theta Functions

“…This can be related to recent results of [11] (see also [12]), where the authors obtain the asymptotic behaviour of a Steinhaus random multiplicative function (basically a multiplicative random variable whose values at prime integers are uniformly distributed on the unit circle). This can be viewed as a random model for .…”

A Note on Helson’s Conjecture on Moments of Random Multiplicative Functions

“…The interested reader should have no trouble working out the details. Theorem 4 has also been obtained independently by Heap and Lindqvist [15].…”