Central limit theorems for random multiplicative functions over function fields

TL;DR

This paper establishes central limit theorems for partial sums of Steinhaus random multiplicative functions over function fields, extending prior work to new settings and providing explicit bounds and inequalities.

Contribution

It provides a characterization for when partial sums of these functions approach a normal distribution in the function field setting, extending recent number field results.

Findings

Central limit theorems for polynomials in short intervals

Results for polynomials with few prime factors

Bounds for rough polynomials in short intervals

Abstract

We provide a sufficient characterization for subsets $\mathcal{A}$ of the polynomial ring $\mathbb{F}_q[t]$ for which partial sums of Steinhaus random multiplicative functions approach a complex standard normal distribution. This extends recent work of Soundararajan and Xu to the function field setting. We apply this characterization to deduce central limit theorems in four cases: polynomials in short intervals, polynomials with few prime factors, shifted primes, and rough polynomials. In doing so, we also establish an explicit Hildebrand inequality for smooth polynomials in short intervals, a function field form of Shiu's theorem for multiplicative functions, and an explicit Chebyshev bound for rough polynomials in short intervals.