Large sets of consecutive Maass forms and fluctuations in the Weyl remainder

TL;DR

This paper introduces a straightforward algorithm combining Hejhal's identity, linearisation, and Turing bounds to compute extensive sets of Maass form eigenvalues, analyzing their fluctuations and proposing related conjectures.

Contribution

It presents a simple, effective algorithm for computing large sets of Maass form eigenvalues and investigates their fluctuation properties, leading to new conjectures.

Findings

Computed over 160,000 eigenvalues of the Laplacian on the modular surface.

Identified statistical properties and asymptotic behavior of Weyl remainder fluctuations.

Formulated conjectures on the maximum size and distribution of the Weyl remainder.

Abstract

We explore an algorithm which systematically finds all discrete eigenvalues of an analytic eigenvalue problem. The algorithm is more simple and elementary as could be expected before. It consists of Hejhal's identity, linearisation, and Turing bounds. Using the algorithm, we compute more than one hundredsixty thousand consecutive eigenvalues of the Laplacian on the modular surface, and investigate the asymptotic and statistic properties of the fluctuations in the Weyl remainder. We summarize the findings in two conjectures. One is on the maximum size of the Weyl remainder, and the other is on the distribution of a suitably scaled version of the Weyl remainder.