Oscillations of random multiplicative functions under initial bias

Abstract

We prove that if $f$ is a random completely multiplicative function, conditional $f(p)=1$ for each prime $p\le (\log x{)}^{2-\hat{I}\mu }$, the probability that ${\sum }_{1\le n\le N}f(n)\ge 0$ for all $N\le x$ is $o(1)$ as $x\to \infty$. This solves a conjecture of Kucheriaviy, who has a complementary result showing this exponent is sharp. We also prove that almost surely the partial sums of $\sum \frac{f(n)}{\sqrt{n}}$ change signs infinitely many times, solving a problem of Aymone.