feat(Wikipedia): asymptotic density of powerful numbers#3715
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joshuasteier wants to merge 1 commit intogoogle-deepmind:mainfrom
Open
feat(Wikipedia): asymptotic density of powerful numbers#3715joshuasteier wants to merge 1 commit intogoogle-deepmind:mainfrom
joshuasteier wants to merge 1 commit intogoogle-deepmind:mainfrom
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mo271
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Apr 12, 2026
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thanks! some initial remarks..
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| def IsPowerful (n : ℕ) : Prop := | ||
| 0 < n ∧ ∀ p ∈ n.primeFactors, p ^ 2 ∣ n |
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Please re-use the already existing defintion for this: see
FormalConjecturesForMathlib/Data/Nat/Full.lean
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| A positive integer $n$ is *powerful* if for every prime $p$ dividing $n$, | ||
| we have $p^2 \mid n$. Equivalently, $n = a^2 b^3$ for some positive integers $a, b$. | ||
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| Let $Q(x) = \#\{ n \leq x : n \text{ is powerful} \}$. Bateman and Grosswald (1958) showed | ||
| $$Q(x) = \frac{\zeta(3/2)}{\zeta(3)} x^{1/2} + \frac{\zeta(2/3)}{\zeta(2)} x^{1/3} + O(x^{1/6}).$$ | ||
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| The leading constant is $\zeta(3/2)/\zeta(3) \approx 2.173$. | ||
| Note: the formula $\zeta(1/2)^{-1} \approx 0.3039$ cited in some sources is incorrect | ||
| ($\zeta(1/2) \approx -1.46$, so its reciprocal is negative). |
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let's avoid duplicating docstrings too much, its fine to keep it brief in the module docstring
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Closes #3686
Formalizes the Bateman–Grosswald (1958) asymptotic Q(x) ~ c·x^(1/2) for powerful numbers.
Note: the issue's formula ζ(1/2)⁻¹ ≈ 0.3039 appears to be a transcription error —
ζ(1/2) ≈ −1.46, so its reciprocal is negative. The correct leading constant from
Bateman–Grosswald is ζ(3/2)/ζ(3) ≈ 2.173, noted in the docstring. Left as
answer(sorry)in the statement; happy to pin a specific form if preferred.