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

Abstract

Let $f(n)$ be a random completely multiplicative function such that $f(p)=\pm 1$ with probabilities $1/2$ independently at each prime. We study the conditional probability, given that $f(p)=1$ for all $p<y$, that all partial sums of $f(n)$ up to $x$ are nonnegative. We prove that for $y\ge C\frac{(\log x{)}^2{\log }_{2}x}{{\log }_{3}x}$ this probability equals $1-o(1)$. We also study the probability $P_x^{\prime }$ that ${\sum }_{n\le x}\frac{f(n)}{n}$ is negative. We prove that $P_x^{\prime }\ll \exp \left(-\exp \left(\frac{\log x{\log }_{4}x}{(1+o(1)){\log }_{3}x}\right)\right)$, which improves a bound given by Kerr and Klurman. Under a conjecture closely related to Halász's theorem, we prove that $P_x^{\prime }\ll \exp (-x^{\hat{I}}\pm )$ for some $\hat{I}\pm >0$. Let $\ddot{I}{‡}_p(n)=\left(\frac{n}{p}\right)$ be the Legendre symbol modulo $p$. For a prime $p$ chosen uniformly at random from $(x,2x]$, we express the probability that all partial sums of $\frac{\ddot{I}{‡}_p(n)}{n}$ are nonnegative in terms of the probability that partial sums of $\frac{f(n)}{n}$ are nonnegative.

32 pages. We replaced the use of Siegel's theorem in the proof of Theorem 4, so the constant there is now effective. We added an 'Outline of the proofs' section and remarks relating the results to subsequent work of Angelo-Xu and Klurman-Mangerel. Minor corrections were made to improve the presentation