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Primes in arithmetic progressions to smooth moduli: A minorant version

Runbo Li

TL;DR

This paper addresses the distribution of primes in arithmetic progressions to smooth moduli by constructing a minorant $\rho(n)$ for the prime indicator $\mathbbm{1}_p(n)$. The author develops a sieve-based decomposition, rooted in Buchstab's identity, to control sums of $\rho(n)$ in residue classes modulo $q$ with $q \mid \prod_{p < x^{\delta}} p$, and proves a dispersion bound with distribution level $\frac{10}{19}$. The result improves on the prior bounds of Baker–Irving and Stadlmann by pushing the admissible modulus exponent to $\frac{10}{19}$ and provides a positive average lower bound for $\rho(n)$, though at the cost of a weaker pointwise lower bound compared to earlier work. The work clarifies the trade-offs between minorant strength and distribution level in the setting of primes in progressions to smooth moduli, with implications for bounded-gap questions under this framework.

Abstract

The author prove that there exists a function $ρ(n)$ which is a minorant for the prime indicator function $\mathbb{1}_{p}(n)$ and has distribution level $\frac{10}{19}$ in arithmetic progressions to smooth moduli. This refines the previous results of Baker--Irving and Stadlmann.

Primes in arithmetic progressions to smooth moduli: A minorant version

TL;DR

This paper addresses the distribution of primes in arithmetic progressions to smooth moduli by constructing a minorant for the prime indicator . The author develops a sieve-based decomposition, rooted in Buchstab's identity, to control sums of in residue classes modulo with , and proves a dispersion bound with distribution level . The result improves on the prior bounds of Baker–Irving and Stadlmann by pushing the admissible modulus exponent to and provides a positive average lower bound for , though at the cost of a weaker pointwise lower bound compared to earlier work. The work clarifies the trade-offs between minorant strength and distribution level in the setting of primes in progressions to smooth moduli, with implications for bounded-gap questions under this framework.

Abstract

The author prove that there exists a function which is a minorant for the prime indicator function and has distribution level in arithmetic progressions to smooth moduli. This refines the previous results of Baker--Irving and Stadlmann.
Paper Structure (3 sections, 4 theorems, 29 equations)

This paper contains 3 sections, 4 theorems, 29 equations.

Table of Contents

  1. Introduction
  2. Asymptotic formulas
  3. The final decomposition

Key Result

Theorem 1.1

There exists a function $\rho(n)$ which satisfies the following properties: (Minorant) $\rho(n)$ is a minorant for the prime indicator function $\mathbbm{1}_{p}(n)$. That is, we have (No small prime factors) If $n$ has a prime factor less than some fixed $\xi > 0$, then $\rho(n) = 0$. (Lower bound) We have (Distribution in Arithmetic Progressions to smooth moduli) For any integer $a$ that coprim

Theorems & Definitions (5)

  • Theorem 1.1
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • Lemma 2.3