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Average first-passage times for character sums

Quanyu Tang, Hao Zhang

Abstract

Let $\varepsilon>0$ and, for an odd prime $p$, set $$ S_\ell(p):=\sum_{n\le \ell}\left(\frac{n}{p}\right). $$ Define the first-passage time $$ f_\varepsilon(p):=\min\{\ell\ge 1:\ S_\ell(p)<\varepsilon\ell\}. $$ We prove that there exists a constant $c_\varepsilon>0$ such that, as $x\to\infty$, $$ \sum_{p\le x} f_\varepsilon(p)\sim c_\varepsilon \frac{x}{\log x}. $$

Average first-passage times for character sums

Abstract

Let and, for an odd prime , set Define the first-passage time We prove that there exists a constant such that, as ,
Paper Structure (9 sections, 10 theorems, 65 equations)

This paper contains 9 sections, 10 theorems, 65 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notation and conventions
  4. Two forms of the quadratic large sieve
  5. Cubic coefficients and a square-condition bound
  6. A density parameter
  7. Proof of Theorem \ref{['thm:intro:first']}
  8. Finite-dimensional equidistribution and limiting tail probabilities
  9. Uniform tail bounds and completion of the proof

Key Result

Theorem 1.2

For every $\varepsilon>0$ there exists a constant $c_\varepsilon\in(0,\infty)$ such that, as $x\to\infty$,

Theorems & Definitions (19)

  • Conjecture 1.1: Erdős's eventual-time problem
  • Theorem 1.2
  • Lemma 2.1: MoVa79
  • Lemma 2.2: HB95
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • proof
  • ...and 9 more