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

Let $f(n)$ be a random completely multiplicative function such that $f(p)=\pm 1$ with probabilities $1\mathrm{/}2$ 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 $P_x^{\mathrm{\prime }}$ that ${\sum }_{n\le x}\frac{f(n)}{n}$ is negative. We prove that $P_x^{\mathrm{\prime }}\ll \exp (-\exp \left(\frac{\log x{\log }_{4}x}{(1+o(1)){\log }_{3}x}\right))$, which improves a bound given by Kerr and Klurman. Under a conjecture closely related to Halász’s theorem, we prove that $P_x^{\mathrm{\prime }}\ll \exp (-x^{\alpha })$ for some $\alpha >0$. Let ${\chi }_p(n)=\left(\frac{n}{p}\right)$ 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 $\frac{{\chi }_p(n)}{n}$ are nonnegative in terms of the probability that partial sums of $\frac{f(n)}{n}$ are nonnegative.