A remark on the average number of divisors of a quadratic polynomial (Q2222866)

In his recent work [Monatsh. Math. 192, 465--478 (2020; Zbl 1455.11129)], using Zagier's determination of the Dirichlet series attached to the function \(d \mapsto \rho_k(4d)\), where \(\rho_k(d)\) counts the number of solutions \(0 \leqslant x < d\) of the congruence \(x^2 \equiv k \pmod d\) [\textit{D. Zagier}, Lect. Notes Math. 627, 105--169 (1977; Zbl 0372.10017)], and also using a non-optimal Tauberian theorem, the author obtained an asymptotic formula for the sum \[\sum_{\frac{1}{2} \kappa < n \leqslant N} \tau(n^2-bn+c)\] \(\tau\) is the usual divisor counting function, \(b,c\) are integers, \(k = b^2-4c :=Df^2\) where \(D\) is a fundamental discriminant and \(\kappa := b\) or \(\kappa := b + \sqrt{k}\) according to whether \(k \leqslant 0\) or \(k > 0\). The main aim of this note is to replace the Tauberian theorem by the work from \textit{C. Hooley} [Acta Math. 110, 97--114 (1963; Zbl 0116.03802)] concerning logarithmic averages of the function \(d \mapsto \rho_k(4d)\) and exponential sums techniques to deal with sum with the first Bernoulli function \(\psi(x) = x - \lfloor x \rfloor - \frac{1}{2}\). This enables him to derive a more precise asymptotic formula for the above sum in the case where \(k\) is not a square, replacing the error term of the form \(O(N)\) of the previous paper by the quantity \(C_k N + O \left( N^{8/9} (\log N)^3 \right)\), where \(C_k\) is an explicit constant depending on \(k\).