Partial sums of typical multiplicative functions over short moving intervals

TL;DR

This paper demonstrates that partial sums of Steinhaus random multiplicative functions exhibit Gaussian behavior over short intervals, with precise conditions on interval length relative to the main variable, advancing understanding of their distribution.

Contribution

It establishes Gaussian limiting distribution for partial sums of typical multiplicative functions over short intervals, extending previous results and addressing a recent open question by Harper.

Findings

Matching moments with Gaussian distribution under specified conditions

Gaussian limit for normalized partial sums in short intervals

Progress on Harper's question about multiplicative functions

Abstract

We prove that the $k$-th positive integer moment of partial sums of Steinhaus random multiplicative functions over the interval $(x, x+H]$ matches the corresponding Gaussian moment, as long as $H\ll x/(\log x)^{2k^2+2+o(1)}$ and $H$ tends to infinity with $x$. We show that properly normalized partial sums of typical multiplicative functions arising from realizations of random multiplicative functions have Gaussian limiting distribution in short moving intervals $(x, x+H]$ with $H\ll X/(\log X)^{W(X)}$ tending to infinity with $X$, where $x$ is uniformly chosen from $\{1,2,\dots, X\}$, and $W(X)$ tends to infinity with $X$ arbitrarily slowly. This makes some initial progress on a recent question of Harper.