Problem 975

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?

Source: erdosproblems.com/975

Number theory
Divisors
Polynomials

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Open

No claim settles this problem.

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

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.

Problem 975

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