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 $1/2$ 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 $P_x'$ that $\sum_{n \le x} \frac{f(n)}{n}$ is negative. We prove that $P_x' \ll \exp \left( - \exp \left( \frac{\log x \log_4 x}{(1 + o(1)) \log_3 x} \right) \right)$, which improves a bound given by Kerr and Klurman. Under a conjecture closely related to Halász's theorem, we prove that $P_x' \ll \exp(-x^α)$ for some $α> 0$. Let $χ_p(n) = \left( \frac{n}{p} \right)$ 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 $\frac{χ_p(n)}{n}$ are nonnegative in terms of the probability that partial sums of $\frac{f(n)}{n}$ are nonnegative.