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Primes in simultaneous arithmetic progressions

Zongkun Zheng

TL;DR

The paper develops a new mean value theorem for the distribution of primes in two simultaneous arithmetic progressions by integrating the dispersion method with spectral theory of Kloosterman sums and a $q$-analogue of van der Corput, drawing on the Bombieri–Friedlander–Iwaniec framework. It provides three complementary dispersion approaches (Type I–III) and leverages Heath–Brown decompositions to bound averages of primes in constrained congruence classes, yielding sharp average results and applications. As a key application, the authors show that there are infinitely many Chen primes $p$ for which $P^+(p+6) > p^{0.217}$, advancing the understanding of shifted prime-factor phenomena and related sieve problems. The work thus supplies new tools for averaging primes in simultaneous arithmetic progressions and has potential implications for prime-tuples and smooth-number analogues in this setting.

Abstract

We prove a new mean value theorem on the distribution of primes in two simultaneous arithmetic progressions. Our approach builds on previous arguments of Bombieri, Fouvry, Friedlander, and Iwaniec appealing to spectral theory of Kloosterman sums, as well as the $q$-analogue of van der Corput method. In particular, we need estimates for exponential sums coming from the spectral theory of automorphic forms (sums of Kloosterman sums) and from algebraic geometry (Weil--Deligne bound for algebraic exponential sums). As an application, we show that the greatest prime factor of $p + 6$ for Chen prime $p$ is infinitely often greater than $p^{0.217}$.

Primes in simultaneous arithmetic progressions

TL;DR

The paper develops a new mean value theorem for the distribution of primes in two simultaneous arithmetic progressions by integrating the dispersion method with spectral theory of Kloosterman sums and a -analogue of van der Corput, drawing on the Bombieri–Friedlander–Iwaniec framework. It provides three complementary dispersion approaches (Type I–III) and leverages Heath–Brown decompositions to bound averages of primes in constrained congruence classes, yielding sharp average results and applications. As a key application, the authors show that there are infinitely many Chen primes for which , advancing the understanding of shifted prime-factor phenomena and related sieve problems. The work thus supplies new tools for averaging primes in simultaneous arithmetic progressions and has potential implications for prime-tuples and smooth-number analogues in this setting.

Abstract

We prove a new mean value theorem on the distribution of primes in two simultaneous arithmetic progressions. Our approach builds on previous arguments of Bombieri, Fouvry, Friedlander, and Iwaniec appealing to spectral theory of Kloosterman sums, as well as the -analogue of van der Corput method. In particular, we need estimates for exponential sums coming from the spectral theory of automorphic forms (sums of Kloosterman sums) and from algebraic geometry (Weil--Deligne bound for algebraic exponential sums). As an application, we show that the greatest prime factor of for Chen prime is infinitely often greater than .
Paper Structure (45 sections, 41 theorems, 316 equations)

This paper contains 45 sections, 41 theorems, 316 equations.

Table of Contents

  1. Introduction
  2. Primes in arithmetic progressions
  3. An application to shifted Chen primes
  4. Notation and preliminaries
  5. Notation and conventions
  6. Combinatorial decompositions of primes
  7. Congruences
  8. Exponential sums
  9. $q$-analogue of van der Corput method
  10. Dispersion method for the general bilinear form
  11. Evaluation of $\mathfrak{S}_3$
  12. Evaluation of $\mathfrak{S}_2$
  13. Preliminary analysis of $\mathfrak{S}_1$
  14. Estimation of $\mathscr{R}_1$: the first method
  15. Preparation for Kloostermania
  16. ...and 30 more sections

Key Result

Theorem 1.1

Let $\varepsilon>0$ and $0\leqslant\theta\leqslant 7/36.$ Suppose $|a_1|\leqslant \log^B x$ for some $B>0,$$|a_2|\leqslant x$ and $a_1\neq a_2$. Let $\boldsymbol{\gamma}=(\gamma_q)$ and $\boldsymbol{\lambda}=(\lambda_d)$ be divisor-bounded sequences supported on positive integers with Assume further that $\boldsymbol{\lambda}$ is well-factorable of level $x^{\mathcal{L}(\theta)-\varepsilon}.$ The

Theorems & Definitions (44)

  • Definition 1.1: Well-factorable
  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Definition 2.1: Siegel--Walfisz condition
  • Definition 2.2: Fourier transform
  • Lemma 2.1: Heath-Brown identity
  • Lemma 2.2
  • Lemma 2.3
  • Lemma 2.4: Bezout's identity
  • ...and 34 more