Sign changes of the partial sums of a random multiplicative function II

2023/03/26 by Aymone, Marco · 1 citation
#FOS: Mathematics #Number Theory (math.NT) #Probability (math.PR)

paper · doi:10.48550/arxiv.2303.14682

Abstract

We study two models of random multiplicative functions: Rademacher random multiplicative functions supported on the squarefree integers f, and Rademacher random completely multiplicative functions f^*. We prove that the partial sums ∑_{n≤ x}f^*(n) and ∑_{n≤ x}(f(n))/(√(n)) change sign infinitely often as x→∞, almost surely. The case ∑_{n≤ x}(f^*(n))/(√(n)) is left as an open question and we stress the possibility of only a finite number of sign changes, with positive probability.

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• Positivity of partial sums of a random multiplicative function and corresponding problems for the Legendre symbol

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