Twisted Correlations of the divisor function via discrete averages of Poincare series

The spectral theory of automorphic forms finds remarkable applications in analytic number theory. Notably, it is utilised in results concerning the distribution of primes in large arithmetic progressions and in questions on variants of the fourth moment of the zeta function. Traditionally, these problems are addressed by reducing them to sums of Kloosterman sums, followed by either the use of existing black-box results or by-hand application of spectral theory through Kuznetsov's formula.In this presentation, based on joint work with Jori Merikoski, I will introduce an alternative approach that entirely circumvents the need for Kloosterman sums. This approach offers increased flexibility compared to existing black-box methods. It is inspired by Bruggeman-Motohashi's work on the fourth moment of the zeta function. As an application, I will present a novel result on correlations of the divisor function in arithmetic progressions.