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.
Categories: math.CA math.AP