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Erdős problem 975 · Conjectures.io
Number theory
Erdős problem 975
For an irreducible polynomial f∈Z[x] with f(n)≥1 for sufficiently large n,
does there exists a constant c=c(f)>0 such that
∑n≤xτ(f(n))≈c⋅xlogx?
Note that it is unclear whether the polynomial should have integer coefficients or merely be
integer-valued. We assume the former.
0 α bounty·nobody has started
week of 11 Oct 2025 · nothing publishedweek of 18 Oct 2025 · nothing publishedweek of 25 Oct 2025 · nothing publishedweek of 1 Nov 2025 · nothing publishedweek of 8 Nov 2025 · nothing publishedweek of 15 Nov 2025 · nothing published
Solve it
Nobody has attempted this. A complete proof or refutation takes the whole bounty.
11 Oct 2025nothing published against this one yet3 Oct 2026
erdosproblems.com/975
[Va39] van der Corput, J. G., Une in\'egalit\'e{} relative au nombre des diviseurs. Nederl. Akad. Wetensch., Proc. (1939), 547--553.
[Er52b] Erd\"os, P., On the sum {∑k=1xd(f(k))}. J. London Math. Soc. (1952), 7--15.
[Ho63] Hooley, Christopher, On the number of divisors of a quadratic polynomial. Acta Math. (1963), 97--114.
[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.
[T] T. Tao, Erdos' divisor bound, https://terrytao.wordpress.com/2011/07/23/erdos-divisor-bound/
Something wrong with this formalization?
A statement that does not faithfully capture the original conjecture is the one real risk here, so we would rather hear about it early - before someone spends weeks on it.
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