Optimal estimates for an average of Hurwitz class numbers

TL;DR

This paper provides an optimal estimate for the average of Hurwitz class numbers and demonstrates their equidistribution in a specific family, leading to new asymptotic relations among these class numbers.

Contribution

It introduces the first optimal estimate for the average of Hurwitz class numbers and applies the resolvent trace formula to establish their equidistribution.

Findings

Optimal estimate for average Hurwitz class numbers

Equidistribution of a specific family weighted by Hurwitz class numbers

Derivation of new asymptotic relations among 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 $\{\frac{t}{2q^{\nu/2}} \ | \ \nu \in \mathbb{N}, t \in \mathbb{Z}, |t|<2q^{\nu/2}\}$ 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 $k\ge 2$.