Moments of polynomials with random multiplicative coefficients

Abstract

For 𝑋(𝑛) a Rademacher or Steinhaus random multiplicative function, we consider the random polynomialsand show that the 2𝑘th moments on the unit circletend to Gaussian moments in the sense of mean-square convergence, uniformly for 𝑘 ≪ (log 𝑁∕ log log 𝑁) 1∕3 , but that in contrast to the case of independent and identically distributed coefficients, this behavior does not persist for 𝑘 much larger. We use these estimates to (i) give a proof of an almost sure Salem-Zygmund type central limit theorem for 𝑃 𝑁 (𝜃), previously obtained in unpublished work of Harper by different methods, and(ii) show that asymptotically almost surelyfor all 𝜀 > 0.M S C 2 0 2 0 11K65 (primary), 11N37, 42A05 (primary)

A note on character sums over short moving intervals

“…However, proving such a result seems difficult. Firstly, the best unconditional estimates we have of the form n≤x χ(n) = o(x), where χ is any non-principal character modulo a prime q (one can sometimes do better for special non-prime moduli), are Burgess-type estimates requiring that x ≥ q 1/4−o (1) . Thus we would need to assume results like GRH merely to exclude the kind of construction from Theorems 1 and 2 from cropping up.…”

“…Probability Result 2 (See Theorem 1.1 of Benatar, Nishry and Rodgers [1]). Let f (n) be an extended Rademacher random multiplicative function.…”

“…where f (k) is a random multiplicative function. This arithmetic input can be extracted from a nice recent paper of Benatar, Nishry and Rodgers [1]. They were interested in almost sure central limit theorems and size bounds for random trigonometric polynomials…”

“…As Benatar, Nishry and Rodgers [1] comment, one can also analyse the distribution of 1 √ N n≤N f (n)e(nθ) using martingale methods, and this was done in unpublished work of the present author (see the paper [9] for an application of martingales to a different distributional problem for random multiplicative functions). But to transfer these conclusions to character sums, one would seem to again need moment estimates on the random multiplicative side, not just distributional convergence.…”

“…These could be obtained (e.g. one can use Burkholder's inequalities [3] and some calculation to show that all moments remain bounded as N → ∞, and this combined with distributional convergence implies they must all converge to the desired Gaussian moments), but it seems simpler to rely on the existing calculations of Benatar, Nishry and Rodgers [1].…”