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A Bombieri-Vinogradov-type theorem for moduli with small radical

Stephan Baier, Sudhir Pujahari

Abstract

In this article, we extend our recent work on a Bombieri-Vinogradov-type theorem for sparse sets of prime powers $p^N\le x^{1/4-\varepsilon}$ with $p\le (\log x)^C$ to sparse sets of moduli $s\le x^{1/3-\varepsilon}$ with radical rad$(s)\le x^{9/40}$. To derive our result, we combine our previous method with a Bombieri-Vinogradov-type theorem for general moduli $s\le x^{9/40}$ obtained by Roger Baker.

A Bombieri-Vinogradov-type theorem for moduli with small radical

Abstract

In this article, we extend our recent work on a Bombieri-Vinogradov-type theorem for sparse sets of prime powers with to sparse sets of moduli with radical rad. To derive our result, we combine our previous method with a Bombieri-Vinogradov-type theorem for general moduli obtained by Roger Baker.
Paper Structure (5 sections, 6 theorems, 80 equations)

This paper contains 5 sections, 6 theorems, 80 equations.

Table of Contents

  1. Introduction and main results
  2. Notations and preliminaries
  3. Proof of Theorem \ref{['mainresult']}
  4. Treatment of type I sums
  5. Treatment of type II sums.

Key Result

Theorem 1

Let $Q\leqslant x^{9/40}$. Let $\mathcal{S}$ be a set of pairwise relatively prime integers in $(Q,2Q]$. Then the number of $q$ in $\mathcal{S}$ for which is $O(\mathcal{L}^{34+A})$.

Theorems & Definitions (6)

  • Theorem 1: Baker
  • Theorem 2: Baier-Pujahari, 2022
  • Theorem 3
  • Proposition 4: Version of Harman's sieve with additional averaging
  • Proposition 5: Large Sieve
  • Proposition 6: Perron's formula