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Primes in arithmetic progressions and short intervals without $L$-functions

Kaisa Matomäki, Jori Merikoski, Joni Teräväinen

Abstract

We develop a sieve that can detect primes in multiplicatively structured sets under certain conditions. We apply it to obtain a new $L$-function free proof of Linnik's problem of bounding the least prime $p$ such that $p\equiv a\pmod q$ (with the bound $p \ll q^{350}$) as well as a new $L$-function free proof that the interval $(x-x^{39/40}, x]$ contains primes for every large $x$. In a future work we will develop the sieve further and provide more applications.

Primes in arithmetic progressions and short intervals without $L$-functions

Abstract

We develop a sieve that can detect primes in multiplicatively structured sets under certain conditions. We apply it to obtain a new -function free proof of Linnik's problem of bounding the least prime such that (with the bound ) as well as a new -function free proof that the interval contains primes for every large . In a future work we will develop the sieve further and provide more applications.
Paper Structure (29 sections, 23 theorems, 175 equations)

This paper contains 29 sections, 23 theorems, 175 equations.

Table of Contents

  1. Introduction
  2. Applications
  3. Linnik's theorem
  4. Primes in short intervals
  5. Sieve theorems
  6. The sieve theorem for finite groups
  7. The sieve theorem for short intervals
  8. Discussion of the proofs
  9. Proof of Theorem \ref{['thm:non-torsion-special']}
  10. Proof of Theorem \ref{['thm:shorts-special']}
  11. Proof of Theorem \ref{['thm_Linnik']}
  12. Proof of Theorem \ref{['thm:Primesinshorts']}
  13. Further developments
  14. Notation and preliminaries
  15. Operations on abelian groups
  16. ...and 14 more sections

Key Result

Theorem 1.1

Let $q\geq q_0$, where $q_0$ is an effectively computable large integer. Then, for every $a \in \mathbb{Z}_q^\times$, there exists a prime $p \leq q^{{350}}$ such that $p \equiv a \ (\mathrm{mod}\ q)$.

Theorems & Definitions (54)

  • Theorem 1.1
  • Theorem 1.2
  • Definition 1.3: Level of distribution
  • Theorem 1.4
  • Remark 1.5
  • Remark 1.6
  • Remark 1.7
  • Theorem 1.8
  • Remark 1.9
  • Remark 1.10
  • ...and 44 more