Problem 975

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claims/: The 4 claim pages of Problem 975, one per claimant's result; the problem's standing derives from them.

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Statement. Let $f\in \mathbb{Z}[x]$ be an irreducible non-constant polynomial such that $f(n)\geq 1$ for all large $n\in\mathbb{N}$. Does there exist a constant $c=c(f)>0$ such that

$$\sum_{n\leq X} \tau(f(n))\sim cX\log X,$$

where $\tau$ is the divisor function?

Status. Open. The site labels the problem OPEN, with its note that no finite computation can settle it (page last edited 27 December 2025).

Source. erdosproblems.com/975, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #975, https://www.erdosproblems.com/975.

References.

• [Er52b] Erdős, P., On the sum $\sum^x_{k=1} d(f(k))$. J. London Math. Soc. 27 (1952), 7-15.
• [Ho63] Hooley, Christopher, On the number of divisors of quadratic polynomials. Acta Math. 110 (1963), 97-114; the site prints the title as "of a quadratic polynomial".
• [Mc95] McKee, James, On the average number of divisors of quadratic polynomials. Math. Proc. Cambridge Philos. Soc. (1995), 389-392.
• [Mc97] McKee, James, A note on the number of divisors of quadratic polynomials. (1997), 275-281.
• [Mc99] McKee, James, The average number of divisors of an irreducible quadratic polynomial. Math. Proc. Cambridge Philos. Soc. (1999), 17-22.
• [Va39] van der Corput, J. G., Une inégalité relative au nombre des diviseurs. Nederl. Akad. Wetensch., Proc. (1939), 547-553.

Formalization. Statement in formal-conjectures. Its erdos_975 states the question, tagged research open; its variant erdos_975.variants.quadratic, tagged research solved and crediting Hooley [Ho63], states the quadratic case with the constant left as answer(sorry), a sorry body and no formal_proof pointer. The file also states the bounds of [Va39] and [Er52b] and the $n^2+1$ asymptotic. A statement file is not a formalization of a result.

Current assessment

The question (site formulation, page last edited 27 December 2025). The statement above; OPEN, with the site's note that no finite computation can settle it. The site's header source is [Er65b], and its commentary credits van der Corput [Va39], Erdős [Er52b], Hooley [Ho63] and McKee [Mc95], [Mc97], [Mc99]. The derived standing is open: every claim page is partial.

Order of magnitude. For every $f$ as in the statement,

$$X\log X\ll_f\sum_{n\le X}\tau(f(n))\ll_fX\log X,$$

the lower bound credited by the site to van der Corput [Va39] and the upper bound Erdős's theorem [Er52b], proved by elementary means; the card for Erdős's paper records the theorem and its lemmas. Erdős's paper says that the lower bound was essentially known, citing Bellman, and that for degree two Bellman and Shapiro had the asymptotic $c\,X\log X+o(X\log X)$, unpublished. Neither bound gets a claim page: they fix the order of the sum for every $f$ but settle the asymptotic for no $f$, so they settle no instance of the question.

The quadratic case. Hooley's Theorem 2 of 1963 proves the asymptotic, with a second main term and the error $O(X^{8/9}\log^3X)$, for every $f(x)=x^2+a$ with $-a$ not a perfect square; it is recorded on his claim page, an accepted partial claim. A monic quadratic $x^2+bx+c$ with even $b$ is a translate of such an $x^2+a$, so the new monic cases have odd $b$. McKee proves the asymptotic with the error $O(x)$ and the constant written in class numbers for every monic irreducible quadratic: for negative discriminant in [Mc95], recorded on his 1995 claim page, and for positive non-square discriminant in [Mc99], recorded on his 1999 claim page, both accepted partial claims. His note [Mc97], in a proceedings volume, treats the non-monic $ax^2+bx+c$ with negative discriminant prime to $a$; it is recorded on its claim page as a claimed partial claim, since no evidence that the volume was refereed is recorded. The non-monic quadratics of positive discriminant, and those of negative discriminant sharing a factor with $a$, have no asymptotic in the sources cited. Scourfield's paper of 1961 (Proc. Glasgow Math. Assoc. 5, 8--20) gets no page: Hooley's introduction credits it with the formula $A_2(a)x\log x+O(x)$ for $x^2+a$, which Hooley's theorem contains, and no review states its theorem. For $f(n)=n^2+1$ the constant is $3/\pi$: $\sum_{n\le x}\tau(n^2+1)=\frac3\pi x\log x+O(x)$, the example the site gives from Tao's blog post on Erdős's bound. The site's discussion thread (16 June 2026) points to Lapkova's paper (arXiv:1704.02498), whose Theorem 1 proves the asymptotic with the constant $2L(1,\chi)/\zeta(2)$ for $n^2+2bn+c$ with squarefree $b^2-c\not\equiv1\pmod4$ and whose Theorem 2 gives an explicit upper bound with the same main term. Lapkova's Theorem 1 gets no page: $n^2+2bn+c=(n+b)^2-(b^2-c)$ is a translate of $x^2+a$ with $-a=b^2-c$ not a square, so it is contained in Hooley's theorem, and its constant agrees with McKee's. Every degree three or more is open: no asymptotic is known for any irreducible $f$ of degree at least three, and Erdős wrote in 1952 that the asymptotic very likely holds for every degree but that he could not prove it.

Search scope, 2026-10-06: the site's problem page (OPEN, last edited 27 December 2025, no proof claims) and its four-comment discussion thread (September 2025 to June 2026); the community database lists the problem as open and as having a formal statement as of its last update; the formal-conjectures statement file is described above.

Linked library material

These entries are derived from explicit links on library pages. They are navigation only and do not by themselves record mathematical progress.

• erdos_1952_sum
• erdos_1952_sum / theorem