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Past events in this series


Thu, 02 May 2024
16:00
Lecture Room 4, Mathematical Institute

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

Jori Merikoski
(University of Oxford)
Abstract

The talk is based on joint work with Lasse Grimmelt. 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 Poincaré series over congruence subgroups while keeping track of interactions between multiple orbits. This approach offers increased flexibility over the widely used sums of Kloosterman sums techniques. We give applications to correlations of the divisor function twisted by periodic functions and the fourth moment of Dirichlet $L$-functions on the critical line.