Abstract:We prove that if $f$ is a random completely multiplicative function, conditional $f(p)=1$ for each prime $p \le (\log x)^{2-\epsilon}$, 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.
From: Rodrigo Angelo [view email]
[v1]
Wed, 6 Nov 2024 23:34:49 UTC (4 KB)
[v2]
Tue, 24 Mar 2026 13:56:32 UTC (11 KB)
[v3]
Sun, 2 Aug 2026 00:55:25 UTC (13 KB)
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