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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
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)2k2+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,,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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