On the exponents of distribution of primes and 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 $x$.

42 pages. Submission arXiv:2404.04239v1 was split into two parts, the first of which is arXiv:2404.04239v2, and the second of which is this submission; added new applications. v2: Corrected a few typos