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Siegel Zeros and the Hardy-Littlewood Conjecture

Yunan Wang

Abstract

In 2016, Fei \cite{fei2016application} established a bound on the Siegel zeros for real primitive Dirichlet characters modulo $q$, assuming the weak Hardy-Littlewood conjecture. Building on Fei's work, Jia \cite{jia2022conditional} demonstrated the same bound using a stronger version of the Hardy-Littlewood conjecture. In this paper, we present a slightly simplified approach to reprove their results.

Siegel Zeros and the Hardy-Littlewood Conjecture

Abstract

In 2016, Fei \cite{fei2016application} established a bound on the Siegel zeros for real primitive Dirichlet characters modulo , assuming the weak Hardy-Littlewood conjecture. Building on Fei's work, Jia \cite{jia2022conditional} demonstrated the same bound using a stronger version of the Hardy-Littlewood conjecture. In this paper, we present a slightly simplified approach to reprove their results.
Paper Structure (6 sections, 5 theorems, 29 equations)

This paper contains 6 sections, 5 theorems, 29 equations.

Table of Contents

  1. Introduction
  2. Some Lemmas
  3. proof of theorem 2
  4. First estimation of srq(x)
  5. Second estimation of rq(x)
  6. Proof of Theorem 4

Key Result

Theorem 2

Let $q$ be a prime number congruent to 3 modulo 4 and let $\chi$ be the real primitive character of modulus $q$. Suppose that the Dirichlet L-function $L(s, \chi)$ has a real exceptional zero $\beta$. If conjecture conj1 is correct then there is a number $c_4>0$ such that for any such $q$ and $\beta

Theorems & Definitions (8)

  • Conjecture 1
  • Theorem 2: Fei(2016)
  • Conjecture 3
  • Theorem 4: Jia(2022)
  • Conjecture 5
  • Theorem 6: Friedlander, Goldston, Iwaniec, and Surajaya(2022)
  • Lemma 7
  • Lemma 8