Primes in arithmetic progressions to large moduli, and shifted primes without large prime factors

TL;DR

This paper proves the infinitude of shifted primes with small prime factors, refines previous bounds, and introduces a new mean value theorem for primes in arithmetic progressions to large moduli, advancing understanding of prime distributions.

Contribution

It presents a new mean value theorem for primes in arithmetic progressions to larger moduli and improves bounds on shifted primes without large prime factors.

Findings

Proved infinitude of shifted primes with prime factors below p^{0.2844}

Extended the range of moduli for primes in arithmetic progressions to x^{17/32}

Improved bounds on the distribution of Carmichael numbers

Abstract

We prove the infinitude of shifted primes $p-1$ without prime factors above $p^{0.2844}$. This refines $p^{0.2961}$ from Baker and Harman in 1998. Consequently, we obtain an improved lower bound on the the distribution of Carmichael numbers. Our main technical result is a new mean value theorem for primes in arithmetic progressions to large moduli. Namely, we estimate primes of size $x$ with quadrilinear forms of moduli up to $x^{17/32}$. This extends moduli beyond $x^{11/21}$, recently obtained by Maynard, improving $x^{29/56}$ from well-known 1986 work of Bombieri, Friedlander, and Iwaniec.