Weyl's Law with Error Estimate

TL;DR

This paper establishes a Weyl's law with an explicit error term for the space of cusp forms on a noncompact symmetric space, using the Selberg trace formula and related techniques.

Contribution

It provides the first precise asymptotic count of cusp forms with an explicit error term for the space X=SL(3,Z)ackslash SL(3,R)/SO(3,R).

Findings

Proves N(λ)=Cλ^(5/2)+O(λ^2) for cusp forms.

Provides an upper bound on cusp forms violating Ramanujan conjecture.

Uses a modified Selberg trace formula and techniques from Stade, Wallace, Huntley, and Tepper.

Abstract

Let X=Sl(3,Z)\Sl(3,R)/SO(3,R). Let N(lambda) denote the dimension of the space of cusp forms with Laplace eigenvalue less than lambda. We prove that N(lambda)=C lambda^(5/2)+O(lambda^2) where C is the appropriate constant establishing Weyl's law with a good error term for the noncompact space X. The proof uses the Selberg trace formula in a form that is modified from the work of Wallace and also draws on results of Stade and Wallace and techniques of Huntley and Tepper. We also, in the course of the proof, give an upper bound on the number of cusp forms that can violate the Ramanujan conjecture.