Revisiting the average number of divisors of a quadratic polynomial

Abstract: 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 $x^{2}-bx+c$ with $b,\,c$ 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.