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A conditional bound for the least prime in an arithmetic progression

Matías Bruna

Abstract

Assuming the generalized Lindelöf hypothesis for Dirichlet $L$-functions, we establish that the least prime $p\equiv a\pmod{q}$ satisfies $p\ll_{\varepsilon} q^{2+\varepsilon}$. This achieves a bound that nearly matches the classical estimate implied by the generalized Riemann hypothesis.

A conditional bound for the least prime in an arithmetic progression

Abstract

Assuming the generalized Lindelöf hypothesis for Dirichlet -functions, we establish that the least prime satisfies . This achieves a bound that nearly matches the classical estimate implied by the generalized Riemann hypothesis.

Paper Structure

This paper contains 9 sections, 11 theorems, 134 equations.

Table of Contents

  1. Introduction
  2. The three principles
  3. Zero-free region
  4. Two zero-density estimates
  5. The Deuring--Heilbronn phenomenon
  6. A prime-detecting sum
  7. Proof of Theorem 1.1
  8. Non-exceptional case
  9. Exceptional case

Key Result

Theorem 1.1

Assume conj-glh. For any $\varepsilon>0$ and any integer $a$ coprime to $q$, there exists a constant $q(\varepsilon)$ such that for all $q\geq q(\varepsilon)$ we have $P(a,q)\leq q^{2+\varepsilon}$. The constant $q(\varepsilon)$ is effectively computable for $\varepsilon>1$ and ineffective otherwise

Theorems & Definitions (20)

  • Conjecture : GLH
  • Theorem 1.1
  • Lemma 2.1: Heath-Brown-Linnik*Lemma 3.2
  • Lemma 2.2
  • proof
  • Proposition 2.3
  • proof
  • Lemma 2.4
  • proof
  • Proposition 2.5
  • ...and 10 more