Oscillations of random multiplicative functions under initial bias

We prove that if $f$ is a random completely multiplicative function, conditional $f(p)=1$ for each prime $p \le (\log x)^{2-ε}$, the probability that $\sum_{1\le n \le N}f(n)\ge 0$ for all $N\le x$ is $o(1)$ as $x \rightarrow \infty$. 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.

💡 Research Summary

The paper studies two fundamental sign‑change problems for a random completely multiplicative function $f:\mathbb N\to{\pm1}$, where each prime $p$ is assigned $f(p)=\pm1$ independently with probability $1/2$, and the values are extended multiplicatively.

Problem 1 – Initial bias.
Fix a parameter $y$ and condition on the event that $f(p)=1$ for every prime $p\le y$. The authors consider the regime $y=o\big((\log x\log\log x)^2\big)$ and ask for the probability that all partial sums $\sum_{n\le N}f(n)$ remain non‑negative for every $N\le x$. Earlier work of Kucheriaviy showed that if $y$ is larger than $C(\log x)^2\log\log x\log\log\log x$ then this probability tends to $1$, and conjectured that the exponent $2$ on $\log x$ is optimal. The present work proves the complementary statement: when $y$ is smaller than the critical scale, the probability that the partial sums stay non‑negative up to $x$ is $o(1)$.

The proof proceeds by analysing the Dirichlet series $F(s)=\sum_{n\le x}f(n)n^{-s}$ at $s=1/2+t$. If the partial sums are non‑negative, Lemma 2.3 shows that $F(s)/s$ is a decreasing function of $s>0$. Using the Euler product, one writes $\log F(s)=R(t)+O(1)$ where
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