On primes in arithmetic progressions and bounded gaps between many primes

TL;DR

This work sharpens our understanding of primes in arithmetic progressions to smooth moduli by proving an exponent of distribution $\theta = \tfrac{1}{2} + \tfrac{1}{40} - \varepsilon$, improving upon prior Polymath bounds. The central advance is a refined $q$-van der Corput process that uses additional averaging over the inner variable to reduce diagonal contributions, enabling a larger admissible range for the key parameters. Combined with Harman’s sieve, this yields an explicit bound $H_m \ll \exp(3.8075\,m)$ for the liminf of prime gaps $p_{n+m}-p_n$, and sharpens small-$m$ bounds for $H_m$ by improving the required admissible $k$-tuples. The results strengthen the link between equidistribution to smooth moduli and bounded prime gaps, and they provide a new toolkit for pushing distribution exponents further in the smooth-modulus regime.

Abstract

We prove that the primes below $x$ are, on average, equidistributed in arithmetic progressions to smooth moduli of size up to $x^{1/2+1/40-ε}$. The exponent of distribution $\tfrac{1}{2} + \tfrac{1}{40}$ improves on a result of Polymath, who had previously obtained the exponent $\tfrac{1}{2} + \tfrac{7}{300}$. As a consequence, we improve results on intervals of bounded length which contain many primes, showing that $\liminf_{n \rightarrow \infty} (p_{n+m}-p_n) = O(\exp(3.8075 m))$. The main new ingredient of our proof is a modification of the q-van der Corput process. It allows us to exploit additional averaging for the exponential sums which appear in the Type I estimates of Polymath.

Theorems & Definitions (66)

• Theorem 1
• Theorem 2
• Corollary 1
• Definition 1: Exponent of distribution
• Definition 2: Coefficient sequences
• Proposition 1
• Lemma 1: Combinatorial Lemma
• proof
• Corollary 2
• proof