Average error for spectral asymptotics on surfaces

Abstract

Let $N(t)$ denote the eigenvalue counting function of the Laplacian on a compact surface of constant nonnegative curvature, with or without boundary. We define a refined asymptotic formula $\widetilde N(t)=At+Bt^{1/2}+C$, where the constants are expressed in terms of the geometry of the surface and its boundary, and consider the average error $A(t)=\frac 1 t \int^t_0 D(s)\,ds$ for $D(t)=N(t)-\widetilde N(t)$. We present a conjecture for the asymptotic behavior of $A(t)$, and study some examples that support the conjecture.