An almost sure upper bound for random multiplicative functions on integers with a large prime factor

2021/05/20 by Mastrostefano, Daniele · 2 citations
#FOS: Mathematics #Number Theory (math.NT) #Primary: 11K65. Secondary: 11N64

paper · doi:10.48550/arxiv.2105.09565

Abstract

Let f be a Rademacher or a Steinhaus random multiplicative function. Let ε>0 small. We prove that, as x→ +∞, we almost surely have |∑_\substackn≤ x
P(n)gt;√(x)f(n)|≤√(x)(loglog x)^1/4+ε, where P(n) stands for the largest prime factor of n. This gives an indication of the almost sure size of the largest fluctuations of f.

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• On the limiting distribution of sums of random multiplicative functions
• A sharp almost sure upper bound for partial sums of random multiplicative functions

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