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Primes in arithmetic progressions to large moduli and refinements of Harman's sieve

Runbo Li

Abstract

We study the average distribution of primes of size $x$ in arithmetic progressions to moduli larger than $x^{\frac{1}{2}}$. Using arithmetic information from the works of many authors together with different variants of the original Harman's sieve, we construct suitable majorants and minorants for the prime indicator function $\mathbbm{1}_{p}(n)$ that satisfy Bombieri--Vinogradov type mean value theorems with different types of moduli. Specifically, we obtain some mean value theorems for primes with bilinear forms of moduli up to $x^{\frac{9}{17}}$ or with trilinear forms of moduli up to $x^{\frac{17}{32}}$.

Primes in arithmetic progressions to large moduli and refinements of Harman's sieve

Abstract

We study the average distribution of primes of size in arithmetic progressions to moduli larger than . Using arithmetic information from the works of many authors together with different variants of the original Harman's sieve, we construct suitable majorants and minorants for the prime indicator function that satisfy Bombieri--Vinogradov type mean value theorems with different types of moduli. Specifically, we obtain some mean value theorems for primes with bilinear forms of moduli up to or with trilinear forms of moduli up to .
Paper Structure (80 sections, 64 theorems, 489 equations)

This paper contains 80 sections, 64 theorems, 489 equations.

Table of Contents

  1. Introduction
  2. General Moduli
  3. Preliminary Lemmas
  4. Type-II estimate
  5. Type-II$_3$ estimate
  6. Type-I/II estimate
  7. Another Type-II estimate
  8. Sieve Asymptotic Formulas
  9. High-dimensional Sieves
  10. Upper Bounds
  11. Case 1. $\frac{1}{2} < \theta < \frac{53}{105}$
  12. Case 2. $\frac{53}{105} \leqslant \theta < \frac{17}{33}$
  13. Case 3. $\frac{17}{33} \leqslant \theta < 0.6$
  14. Case 4. $0.6 \leqslant \theta < 1$
  15. Lower Bounds
  16. ...and 65 more sections

Key Result

Theorem 1.1

Let $Q_1 = x^{\theta_1}$ and $Q_2 = x^{\theta_2}$. Suppose that $\theta_1$ and $\theta_2$ satisfy any of the following conditions: (1). $\frac{1}{4} \leqslant \theta_1 < \frac{3}{10},\ \frac{1 - 2 \theta_1}{2} \leqslant \theta_2 < \min\left(\frac{3 - 4 \theta_1}{8}, \frac{5 - 14 \theta_1}{4} \right)

Theorems & Definitions (87)

  • Theorem 1.1
  • Theorem 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Lemma 2.4
  • Lemma 2.5
  • Lemma 2.6
  • Lemma 2.7
  • Lemma 2.8
  • Lemma 2.9
  • ...and 77 more