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Number theory
Date added
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.
$1,401 bounty·nobody has started
week of 18 Sept 2025 · nothing publishedweek of 25 Sept 2025 · nothing publishedweek of 2 Oct 2025 · nothing publishedweek of 9 Oct 2025 · nothing publishedweek of 16 Oct 2025 · nothing publishedweek of 23 Oct 2025 · nothing published
Solve it
Nobody has attempted this. A complete proof or refutation takes the whole bounty.
18 Sept 2025nothing published against this one yet10 Sept 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.
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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