Average Error for Spectral Asymptotics on Surfaces

Let $N(t)$ denote the eigenvalue counting funtion of the Laplacian on a compact surface of constant nonnegative curvature, with or without boundary. We define a refined asymptotic formula $\tilde{N}(t)=At+Bt^{1/2}+C$, where the constants are expressed in terms of the geometry of the surface and its boundary, and consider the average error $A(t) = \frac{1}{t} \int_0^{t}D(s)ds$ for $D(t) = N(t) - \tilde{N}(t)$. We present a conjecture for the asymptotic behavior of $A(t)$, and study some examples that support the conjecture.