On the exponents of distribution of primes and smooth numbers

TL;DR

This paper proves that primes and smooth numbers are evenly distributed in arithmetic progressions up to a certain large modulus, removing reliance on a major conjecture and providing improved bounds for twin primes and smooth numbers.

Contribution

It establishes distribution results for primes and smooth numbers without depending on Selberg's eigenvalue conjecture, using advanced sieve weights and recent large sieve inequalities.

Findings

Primes and smooth numbers are equidistributed in arithmetic progressions up to x^{5/8 - o(1)}.

Eliminates the need for Selberg's eigenvalue conjecture in related distribution results.

Provides refined upper bounds for twin primes and consecutive smooth numbers.

Abstract

We show that both primes and smooth numbers are equidistributed in arithmetic progressions to moduli up to $x^{5/8 - o(1)}$, using triply-well-factorable weights for the primes (we also get improvements for the well-factorable linear sieve weights). This completely eliminates the dependency on Selberg's eigenvalue conjecture in previous works of Lichtman and the author, which built in turn on results of Maynard and Drappeau. We rely on recent large sieve inequalities for exceptional Maass forms of the author for additively-structured sequences, and on a related result of Watt for multiplicatively-structured sequences. As applications, we prove refined upper bounds for the counts of twin primes and consecutive smooth numbers up to $x$.