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Consider the divisor sum  $$\sum _{n\le N}\tau (n^2+2bn+c)$$ ∑ n ≤ N τ ( n 2 + 2 b n + c ) for integers  b  and  c . We improve the explicit upper bound of this average divisor sum in certain cases, and as an application, we give an improvement in the maximal possible number of  $$D(-1)$$ D ( - 1 ) -quadruples. The new tool is a numerically explicit Pólya–Vinogradov inequality, which has not been formulated explicitly before but is essentially due to Frolenkov–Soundararajan.

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