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 \(k\geqslant 2.\)

中文翻译：

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Hurwitz班级平均数的最佳估计

在本文中，我们给出了Hurwitz类平均数的最佳估计。作为应用，我们给出{{\ mathbb {N}}}中\（\ Big \ {\ frac {t} {2q ^ {\ nu / 2}} \ | \ \ nu \族的均值分布结果， t \ in {{\ mathbb {Z}}}，| t | \ leqslant 2q ^ {\ nu / 2} \ Big \} \）中，带有q质数，由Hurwitz类编号加权。这种均分布在Hurwitz类数之间产生许多渐近关系。我们的证明依赖于权重为\（k \ geqslant 2. \）的椭圆形尖点形式的Hecke算子的解析轨迹公式。