Explicit upper bound for the average number of divisors of irreducible quadratic polynomials

Consider the divisor sum ∑ n ≤ N τ ( n 2 + 2 b n + c ) for integers b and c. We extract an asymptotic formula for the average divisor sum in a convenient form, and provide an explicit upper bound for this sum with the correct main term. As an application we give an improvement of the maximal possible number of D ( - 1 ) -quadruples.