Oscillations of random multiplicative functions under initial bias
Abstract
Abstract We prove that if $f$ is a random completely multiplicative function, conditional on $f(p)=1$ 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.
Fields of Study & Subjects
Coding theory and cryptography
Mathematical Approximation and Integration
Publication details
- Journal / source
- International Mathematics Research Notices
- Volume
- 2026
- Issue
- 18
- Publication date
- 21 August 2026
- Language
- EN