How negative can ∑_{n≤ x}(f(n))/(n) be?

2022/11/10 by Kerr, Bryce, Klurman, Oleksiy
#FOS: Mathematics #Number Theory (math.NT) #Probability (math.PR)

paper · doi:10.48550/arxiv.2211.05540

Abstract

Turán observed that logarithmic partial sums ∑_{n≤ x}(f(n))/(n) of completely multiplicative functions (in the particular case of the Liouville function f(n)=λ(n)) tend to be positive. We develop a general approach to prove two results aiming to explain this phenomena. Firstly, we show that for every ε>0 there exists some x₀≥ 1, such that for any completely multiplicative function f satisfying -1≤ f(n)≤ 1, we have ∑_{n≤ x}(f(n))/(n)≥ -\frac1(loglogx)^1-ε, x≥ x₀. This improves a previous bound due to Granville and Soundararajan. Secondly, we show that if f is a typical (random) completely multiplicative function f:ℕ→ \-1,1\, the probability that ∑_{n≤ x}(f(n))/(n) is negative for a given large x, is O(exp(-exp((log x⋅ logloglog x)/(Clog log x)))). This improves on recent work of Angelo and Xu.

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