The average number of divisors of an irreducible quadratic polynomial

Abstract

For a non-zero integer n, let d(n) denote the number of positive divisors of n. Let a, b and c be integers with a>0, and set Δ=b²−4ac. If the quadratic polynomial ax²+bx+c is irreducible over the rational numbers Q (that is, if Δ is not the square of an integer), then one has

formula here

as X→∞, for some λ depending on a, b and c (see [7]). In this paper we discuss the way in which λ depends on a, b and c, giving a precise, compact expression in terms of class numbers. This extends previous work for the case a=1, Δ<0 (see [4]).

For the case a=1, b=0, a much better description of the error is given in [2], with the following expression for λ:

formula here

Here ρ is a multiplicative function, defined below, and (p/q) is the Legendre/Jacobi symbol.