On a Turán conjecture and random multiplicative functions

2022/05/12 by Rodrigo Angelo, Angelo, Rodrigo, Max Wenqiang Xu +1 · 2 citations
Mathematics · #Analytic Number Theory Research #FOS: Mathematics #Limits and Structures in Graph Theory #Mathematical Dynamics and Fractals #Number Theory (math.NT) #Probability (math.PR)

paper · pdf · doi:10.48550/arxiv.2205.05822

openalex publication_date 2022/05/12 · openalex created_date 2022/05/22 · openalex updated_date 2026/07/28

Abstract

We show that if f is the random completely multiplicative function, the probability that ∑_{n≤ x}(f(n))/(n) is positive for every x is at least 1-10^-45, while also strictly smaller than 1. For large x, we prove an asymptotic upper bound of O(exp(-exp( (log x)/(Clog log x )))) on the exceptional probability that a particular truncation is negative, where C is some positive constant.

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

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