Distribution of the ternary divisor function in arithmetic progressions

Abstract: The ternary divisor function, denoted $\tau_3(n)$, counts the number of ways to write a natural number $n$ as an ordered product of three positive integers. Thus, $\sum_{n=1}^\infty \tau_3(n) n^{-s} = \zeta^3(s).$ Given two coprime positive integers $a$ and $q$, we study the distribution of $\tau_3$ in arithmetic progressions $n \equiv a (\text{mod } q).$ The distribution of $\tau_3$ in arithmetic progressions has a rich history and has applications to the distribution of prime numbers and moments of Dirichlet $L$-functions. We show that $\tau_3$ is equidistributed on average for moduli $q$ up to $X^{2/3}$, extending the individual estimate of Friedlander and Iwaniec (1985). We will also discuss an averaged variance of $\tau_3$ in arithmetic progressions related to a recent conjecture of Rodgers and Soundararajan (2018) about asymptotic of this variance. One of the key inputs to this asymptotic come from a modified additive correlation sum of $\tau_3.$

algebraic geometrynumber theory

Audience: researchers in the topic

UCSB Seminar on Geometry and Arithmetic