Partial sums of typical multiplicative functions over short moving intervals

Pandey, Mayank; Wang, Victor Y.; Xu, Max Wenqiang

doi:10.2140/ant.2024.18.389

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Partial sums of typical multiplicative functions over short moving intervals

Mayank Pandey, Victor Y. Wang and Max Wenqiang Xu

Vol. 18 (2024), No. 2, 389–408

DOI: 10.2140/ant.2024.18.389

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 $H ≪ x ∕ {(\log x)}^{2 k^{2} + 2 + o (1)}$ 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 $H ≪ X ∕ {(\log X)}^{W (X)}$ 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.

Keywords

random multiplicative function, short moving intervals, multiplicative Diophantine equations, paucity, Gaussian behavior, correlations of divisor functions

Mathematical Subject Classification

Primary: 11K65

Secondary: 11D45, 11D57, 11D79, 11N37

Milestones

Received: 24 July 2022

Revised: 8 March 2023

Accepted: 13 May 2023

Published: 6 February 2024

Authors

Mayank Pandey
Princeton University Princeton, NJ United States

Victor Y. Wang
Courant Institute New York University New York, NY United States

Max Wenqiang Xu
Department of Mathematics Stanford University Stanford, CA United States

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Vol. 18, No. 2, 2024