We prove a congruence modulo 16 relating the class numbers $h(-4p)$, $h(16p)$ of quadratic orders and Zagier's sum $m(4p)$ associated to $\sqrt{4p}$, when $p\equiv 1$ (mod 4) is a prime. This gives an analogy to Chua-Gunby-Park-Yuan's congruence established when $p\equiv 3$ (mod 4), and generalizes a recent work by Cheng and Guo. In particular, when $p\equiv 1$ (mod 4) is a prime, it is shown that the class number $h(-4p)$ is divisible by 16 if and only if the Zagier sum $m(4p)$ is divisible by 16.

中文翻译：

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与二次阶的类数和Zagier的和相关的同余

我们证明与类编号相关的全模16 $H（-4p）$， $H（16p）$ 二次阶和Zagier的总和 $米（4p）$ 与...相关 $\sqrt{4p}$， 什么时候 $p\equiv \mathrm{1个}$（mod 4）是素数。这可比喻为蔡-冈比-帕克-袁（Chua-Gunby-Park-Yuan）在$p\equiv 3$（第4版），并概括了Cheng和Guo的最新著作。特别是什么时候$p\equiv \mathrm{1个}$ （mod 4）是素数，它表明班级号 $H（-4p）$ 当且仅当Zagier总和可被16整除 $米（4p）$ 被16整除