Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Sign changes of the partial sums of a random multiplicative function II

Marco Aymone

Abstract

We study two models of random multiplicative functions: Rademacher random multiplicative functions supported on the squarefree integers $f$, and Rademacher random completely multiplicative functions $f^*$. We prove that the partial sums $\sum_{n\leq x}f^*(n)$ and $\sum_{n\leq x}\frac{f(n)}{\sqrt{n}}$ change sign infinitely often as $x\to\infty$, almost surely. The case $\sum_{n\leq x}\frac{f^*(n)}{\sqrt{n}}$ is left as an open question and we stress the possibility of only a finite number of sign changes, with positive probability.

Sign changes of the partial sums of a random multiplicative function II

Abstract

We study two models of random multiplicative functions: Rademacher random multiplicative functions supported on the squarefree integers , and Rademacher random completely multiplicative functions . We prove that the partial sums and change sign infinitely often as , almost surely. The case is left as an open question and we stress the possibility of only a finite number of sign changes, with positive probability.
Paper Structure (11 sections, 5 theorems, 25 equations)

This paper contains 11 sections, 5 theorems, 25 equations.

Table of Contents

  1. Introduction
  2. Main results and motivation
  3. Proof method
  4. Background
  5. Preliminaries
  6. Notation
  7. Some Lemmas
  8. Proof of the main results
  9. The case $\sum_{n\leq x}\frac{f(n)}{\sqrt{n}}$
  10. The case $\sum_{n\leq x}\frac{f^*(n)}{n^\alpha}$
  11. The case $\sum_{n\leq x}\frac{f(n)}{n^\alpha}$

Key Result

Theorem 1.1

Let $f$ be a Rademacher random multiplicative function. Then for each $0\leq\alpha\leq 1/2$, $\sum_{n\leq x}\frac{f(n)}{n^\alpha}$ changes sign infinitely often as $x\to\infty$, almost surely.

Theorems & Definitions (5)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3: Harper harpergaussian, page 25
  • Lemma 2.1
  • Lemma 2.2