Twisted correlations of the divisor function via discrete averages of $\operatorname{SL}_2(\mathbb{R})$ Poincaré series

Lasse Grimmelt, Jori Merikoski

Published 2024-04-12Version 1

We prove a theorem that allows one to count solutions to determinant equations twisted by a periodic weight with high uniformity in the modulus. It is obtained by using spectral methods of $\operatorname{SL}_2(\mathbb{R})$ automorphic forms to study Poincar\'e series over congruence subgroups while keeping track of interactions between multiple orbits. In this way we get advantages over the widely used sums of Kloosterman sums techniques. We showcase the method with applications to correlations of the divisor functions twisted by periodic functions and the fourth moment of Dirichlet $L$-functions on the critical line.