Optimal estimates for an average of Hurwitz class numbers

Abstract

In this paper, we give an optimal estimate of an average of Hurwitz class numbers. As an application, we give an equidistribution result of the family \(\Big \{\frac{t}{2q^{\nu /2}} \ | \ \nu \in {{\mathbb {N}}}, t \in {{\mathbb {Z}}}, |t|\leqslant 2q^{\nu /2}\Big \}\) with q prime, weighted by Hurwitz class numbers. This equidistribution produces many asymptotic relations among Hurwitz class numbers. Our proof relies on the resolvent trace formula of Hecke operators on elliptic cusp forms of weight \(k\geqslant 2.\)