Kuznecov remainders and generic metrics

TL;DR

This paper improves the remainder estimates in the Kuznecov sum formula for Laplace eigenfunctions on Riemannian manifolds, showing that for generic metrics, the oscillatory second term can be removed, leading to sharper bounds.

Contribution

It provides new remainder estimates in the Kuznecov formula and demonstrates that for a generic class of metrics, the second oscillatory term can be eliminated.

Findings

Improved remainder estimates in the Kuznecov sum formula.

For generic metrics, the oscillatory second term can be eliminated.

Sharper bounds on period integrals of Laplace eigenfunctions.

Abstract

We obtain improved remainder estimates in the Kuznecov sum formula for period integrals of Laplace eigenfunctions on a Riemannian manifold $M$. Building upon the two-term asymptotic expansion established in [arXiv:2204.13525], we prove that for a Baire-generic class of metrics, the oscillatory second term in the Kuznecov formula can be eliminated, yielding an improved remainder estimate.