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Random multiplicative functions and typical size of character in short intervals

Rachid Caich

Abstract

We examine the conditions under which the sum of random multiplicative functions in short intervals, given by $\sum_{x<n \leqslant x+y} f(n)$, exhibits the phenomenon of \textit{better than square-root cancellation}. We establish that the point at which the square-root cancellation diminishes significantly is approximately when the ratio $\log\big(\frac{x}{y}\big)$ is around $\sqrt{\log\log x}$. By modeling characters by random multiplicative functions, we give a sharp bound of $\frac{1}{r-1}\sum_{χ\!\!\!\mod r} \big|\sum_{x<n\leqslant x+y}χ(n)\big|$, where $r$ is a large prime and $x+y\leqslant r $. This extends the result of Harper \cite{Harper_charac}.

Random multiplicative functions and typical size of character in short intervals

Abstract

We examine the conditions under which the sum of random multiplicative functions in short intervals, given by , exhibits the phenomenon of \textit{better than square-root cancellation}. We establish that the point at which the square-root cancellation diminishes significantly is approximately when the ratio is around . By modeling characters by random multiplicative functions, we give a sharp bound of , where is a large prime and . This extends the result of Harper \cite{Harper_charac}.
Paper Structure (17 sections, 24 theorems, 241 equations)

This paper contains 17 sections, 24 theorems, 241 equations.

Table of Contents

  1. Introduction
  2. Sketch of the proof
  3. Preliminary results
  4. Some known results
  5. Reduction of the proof
  6. The case of random multiplicative functions
  7. The proof of the upper bound.
  8. Intermediate upper bound.
  9. Multiplicative Chaos
  10. The proof of the lower bound.
  11. The case of character sums
  12. Preliminaries
  13. Proof of Theorem \ref{['theoreme_principal_3']}.
  14. The conditioning argument
  15. Passing to the random multiplicative functions
  16. ...and 2 more sections

Key Result

Theorem 1.1

Let $f$ be a Rademacher or Steinhaus random multiplicative function. Let $y\leqslant x$ large numbers, we define $\theta$ by $\frac{x}{y}=\log^{\theta} x$. We have uniformly for $\theta > 0$ and $0\leqslant q \leqslant 1$ In particular for $\theta = o(\frac{1}{\sqrt{\log_2 x}})$, we have

Theorems & Definitions (51)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Remark 1
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • proof
  • ...and 41 more