Revisiting the average number of divisors of a quadratic polynomial (Q2187966)

The author proves an asymptotic for the average number of divisors of the quadratic polynomial \(x^2+bx+c\). The problem has been studied extensively in the literature. \textit{J. McKee} in the two papers [Math. Proc. Camb. Philos. Soc. 117, No. 3, 389--392 (1995; Zbl 0841.11050); ibid. 126, No. 1, 17--22 (1999; Zbl 0923.11132)] provides a treatment for the case of irreducible quadratics and he shows an explicit expression \[\sum_{n\leq X} \tau(x^2+bx+c)=\lambda X\log(X)+O(X),\] where \(\lambda\) is related to the Hurwitz class number for \(\Delta<0\) and to the narrow class group if \(\Delta>0\). \textit{C. Hooley} [Acta Math. 110, 97--114 (1963; Zbl 0116.03802)] and \textit{K. Lapkova} [J. Number Theory 180, 710--729 (2017; Zbl 1421.11078)] obtained an asymptotic formula for special cases of reducible quadratic polynomials; Hooley for polynomial of the form \(x^2-r^2\) and Lapkova for polynomials of the form \((x-b)(x-c)\) where \(b<c\) have the same parity. The goal of the author is to provide an uniform treatment for any quadratic polynomial using a well-known result of \textit{D. Zagier} [Lect. Notes Math. 627, 105--169 (1977; Zbl 0372.10017)]. As a byproduct of the formula the author obtains 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.