The Landau-Selberg-Delange method for products of Dirichlet $L$-functions, and applications, I

Abstract

The Landau-Selberg-Delange method gives precise asymptotic formulas for the partial sums $\sum_{n \le x} \, a_n$ of a Dirichlet series $\sum_n \, a_n/n^s$ that behaves like a complex power of the Riemann zeta function. However, situations often arise when the Dirichlet series behaves like a product of complex powers of several Dirichlet $L$-functions to a modulus $q$. In such situations, one often requires sharp asymptotic formulas for the partial sums $\sum_{n \le x} \, a_n$ that apply in much wider ranges of $q$ than permitted by known forms of the Landau-Selberg-Delange method. In this manuscript, we address this problem, giving new estimates on $\sum_{n \le x} \, a_n$ in ranges of $q$ that are (in most applications) much wider than attainable from previous results. Our results also weaken certain hypotheses on the size of $\{a_n\}_n$. As applications of our main theorems, we extend Landau's classical results on the distribution of integers with prime factors restricted to progressions, and improve upon Chang, Martin and Nguyen's work on the distributions of the least invariant factors and least primary factors of multiplicative groups. We also extend the classical Sathe-Selberg theorem and study the local laws of the functions $\Omega_a(n)$ and $\omega_a(n)$, that count (with and without multiplicity, respectively), the number of prime divisors of $n$ lying in the progression $a$ mod $q$.