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Claim. Let $f(n)=an^2+bn+c$ with $\Delta=b^2-4ac<0$ and $\gcd(a,\Delta)=1$. Then

$$\sum_{n\le x}d(f(n))=C\,x\log x+O(x),$$

where $d$ is the divisor function and the constant $C$ is given explicitly in terms of $\Delta$ and $a$; this is the theorem as the zbMATH review of the paper (Zbl 0935.11034) states it. The proof studies the sum $\sum_{k\le x}\rho(k)$, where $\rho(k)$ counts the roots of $f(n)\equiv0\pmod k$ with $0\le n<k$, by an approach different from the one of [[problems/polynomials/E0975/claims/1995_05_01_mckee|McKee's 1995 paper]]. In the notation of Problem 975, for such $f$ with $a>0$ the asymptotic of the problem holds with $c(f)=C$, positive because the sum is $\gg X\log X$.

Covers. The quadratics $ax^2+bx+c$ with $a>0$, $b^2-4ac<0$ and $\gcd(a,b^2-4ac)=1$; the new instances are those with $a\ge2$, such as $2x^2+x+1$. Not covered: the non-monic quadratics of negative discriminant sharing a factor with $a$, the non-monic quadratics of positive non-square discriminant, and every degree three or more.

Depends on. No page of this wiki.

Acceptance. J. McKee, A note on the number of divisors of quadratic polynomials, in Sieve methods, exponential sums, and their applications in number theory (Cardiff, 1995), London Math. Soc. Lecture Note Ser. 237, Cambridge Univ. Press, 1997, 275--281, a proceedings volume; the record dates it to 30 January 1997. No evidence that the volume was refereed is recorded, so no refereed evidence is listed. The curator of erdosproblems.com, Thomas Bloom, cites this paper as [Mc97] in the problem's commentary, but the site labels the problem OPEN, so that citation is not acceptance.