A remark on the average number of divisors of a quadratic polynomial

Abstract

In recent work, we use Dudek’s method together with a result of Zagier to establish an asymptotic formula for the average number of divisors of an irreducible quadratic polynomial of the form $$x^{2}-bx+c$$ with b, c integers. In this note, we remark that one can adopt the work of Hooley to derive a more precise asymptotic formula for the case $$x^{2}-bx+c$$ with $$b^{2}-4c$$ not a square, and as a consequence, re-establish the weaker asymptotic formula given in our recent work by different arguments.