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Claim. Theorem 2 of the paper: let $a$ be a nonzero integer such that $-a$ is not a perfect square. Then, as $x\to\infty$,

$$\sum_{\substack{n\le x\\ n^2+a>0}}d(n^2+a) =A_2(a)\,x\log x+A_3(a)\,x+O\bigl(x^{8/9}\log^3x\bigr),$$

where $d$ is the divisor function and $A_2(a)$ and $A_3(a)$ are explicit constants depending on $a$: $A_2(a)=2f_{-a}(1)$ for a Dirichlet series $f_{-a}(s)$ that the theorem writes in terms of $\zeta(s)/\zeta(2s)$ and the $L$-functions of the quadratic characters attached to $-a$, and $A_3(a)$ involves $f_{-a}(1)$, $f'_{-a}(1)$ and Euler's constant. In the notation of Problem 975, for $f(x)=x^2+a$ with $-a$ not a perfect square, which is exactly the condition that $f$ be irreducible over $\mathbb Z$, and for which $f(n)\ge1$ for all large $n$,

$$\sum_{n\le X}\tau(f(n))\sim c(f)\,X\log X,\qquad c(f)=A_2(a)>0,$$

the constant being positive because the sum is $\gg X\log X$. The proof writes the sum as three sums, two of which give the main terms, and bounds the third through a new exponential sum over the roots of the congruence $\nu^2\equiv-a\pmod k$, estimated with the theory of binary quadratic forms; Section 9 derives from the same estimate that the roots $\nu/k$ are uniformly distributed (Theorem 3). The introduction says that the case $-a=k^2$ is excluded because $n^2-k^2$ factors and the sum then has order $x\log^2x$, that the elementary formula $A_2(a)x\log x+O(x)$ had been commonly realized and proved by Scourfield with a weaker error term, and that similar but more complicated methods give the asymptotic for $\sum d(an^2+bn+c)$; the paper proves it only for $n^2+a$.

Covers. The quadratics $f(x)=x^2+a$ with $a\ne0$ and $-a$ not a perfect square: for every such $f$ the asymptotic of the problem holds with $c(f)=A_2(a)$. Not covered: the other irreducible quadratics, for which McKee proves the asymptotic with the constant written in class numbers, for the monic ones in 1995 and 1999 and for some non-monic ones in 1997; and every degree three or more, where only the order of magnitude $X\log X$ is known and the problem is open.

Depends on. No page of this wiki.

Acceptance. C. Hooley, On the number of divisors of quadratic polynomials, Acta Math. 110 (1963), 97--114, received 5 April 1963, a refereed journal (refereed); the record gives only the year, so the page is dated to its first day. The site's reference [Ho63] prints the title with "a quadratic polynomial"; the record's title is the plural. The curator of erdosproblems.com, Thomas Bloom, credits the quadratic case to this paper in the problem's commentary, but the site labels the problem OPEN, so that credit is not acceptance of this partial claim. Lapkova (arXiv:1704.02498, Section 1) restates Hooley's constant for $x^2+C$ and McKee's for the monic irreducible quadratic.