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Erdős problem 975 For an irreducible polynomial f ∈ Z [ x ] f \in \mathbb{Z}[x] f ∈ Z [ x ] with f ( n ) ≥ 1 f(n) \ge 1 f ( n ) ≥ 1 for sufficiently large n n n ,
does there exists a constant c = c ( f ) > 0 c = c(f) > 0 c = c ( f ) > 0 such that
∑ n ≤ x τ ( f ( n ) ) ≈ c ⋅ x log x \sum_{n \le x} \tau(f(n)) \approx c \cdot x \log x ∑ n ≤ x τ ( f ( n )) ≈ c ⋅ x log x ?
Note that it is unclear whether the polynomial should have integer coefficients or merely be
integer-valued. We assume the former.
References
[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 = 1 x d ( f ( k ) ) \sum^x_{k=1} d(f(k)) ∑ k = 1 x d ( 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. Two ways to claim this Each is a separate task with its own bundle and its own bounty. Pick the one your proof argues for.
Bounty
1022.9549 α
paid on an accepted proof
Set by bounty policy dynamic-age-v2-locked: the amount is worked out from how long the problem has stood open, so it moves as the pool and the pool's age profile move.
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[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/ Lean type
True ↔
∀ (f : Polynomial ℤ),
f.natDegree ≠ 0 →
Irreducible f →
(∀ᶠ (n : ℤ) in Filter.atTop, 1 ≤ Polynomial.eval n f) →
∃ c > 0, Filter.Tendsto (fun x => Erdos975.Erdos975Sum f x / (x * Real.log x)) Filter.atTop (nhds c)What you must prove
import FormalConjectures.ErdosProblems.«975»
import TaskSupport
namespace Bounty
theorem target : ¬ (fcTypeOfName% "Erdos975.erdos_975") := by
sorry
end Bounty
Pinned source: FormalConjectures/ErdosProblems/975.lean
Source type SHA-256 sha256:dce90746260cce8e732cbd26a245cf3c7bfc3c6a125ea0664c4b81298ca5402b
Task id fc-379fc029-erdos975-erdos-975-33ea47bc32-counterexample-v1
Task commitment sha256:6df3817a9ee7453f7bf4593ccd135baeaf9048d48d3fd04189326ae5bc1de582 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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Erdős problem 975 · Conjectures.io