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Oscillations of random multiplicative functions under initial bias

Rodrigo Angelo, Max Wenqiang Xu

Abstract

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.

Oscillations of random multiplicative functions under initial bias

Abstract

We prove that if is a random completely multiplicative function, conditional for each prime , the probability that for all is as . 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 change signs infinitely many times, solving a problem of Aymone.

Paper Structure

This paper contains 3 sections, 7 theorems, 26 equations.

Table of Contents

  1. Introduction
  2. Proof of Theorem \ref{['maintheorem']}
  3. Proof of Theorem \ref{['thm: sign changes']}

Key Result

Theorem 1.1

Let $f$ be the random completely multiplicative function. Let $x$ be large and suppose $y = o\left(\left(\frac{\log x}{\log \log x}\right)^2\right)$. Then, conditional on $f(p)=1$ for each $p \le y$, the probability that $\sum_{n \le N} f(n) \ge 0$ for each $N \le x$ is $o(1)$ as $x \rightarrow \inf

Theorems & Definitions (15)

  • Theorem 1.1
  • Theorem 1.2
  • Lemma 2.1
  • proof : Proof of Lemma \ref{['centrallimit']}
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • ...and 5 more