In this work, we employ a well known result due to Zagier to derive an asymptotic formula for the average number of divisors of a quadratic polynomial of the form $$x^{2}-bx+c$$ x 2 - b x + c with $$b,\,c$$ b , c integers. As simple consequences, we obtain formulas for computing the narrow class number of a quadratic field and relations between the narrow class number and a weighted class number for binary quadratic forms considered by McKee. In the end, we employ the class number relation to compute several examples of the weighted class number.

中文翻译：

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重新审视二次多项式的平均除数

在这项工作中，我们采用了 Zagier 的众所周知的结果来推导出 $$x^{2}-bx+c$$ x 2 - bx + c 形式的二次多项式的平均除数的渐近公式用 $$b,\,c$$ b , c 整数。作为简单的结果，我们获得了计算二次域的窄类数的公式以及 McKee 考虑的二进制二次形式的窄类数和加权类数之间的关系。最后，我们使用类数关系来计算加权类数的几个例子。