Consider the divisor sum $\sum_{n\leq N}\tau(n^2+2bn+c)$ for integers $b$ and $c$ which satisfy certain extra conditions. For this average sum we obtain an explicit upper bound, which is close to the optimal. As an application we improve the maximal possible number of $D(-1)$-quadruples.
Explicit upper bound for the average number of divisors of irreducible quadratic polynomials
An explicit upper bound for the least prime ideal in the Chebotarev density theorem
An explicit upper bound for $L(1, χ)$ when $χ$ is quadratic
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