feat(IwaniecKowalskiCh1): prove liouville_eq_moebius_on_squarefree#1482
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feat(IwaniecKowalskiCh1): prove liouville_eq_moebius_on_squarefree#1482foolishair wants to merge 1 commit into
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For squarefree n, both λ(n) and μ(n) equal (-1)^Ω(n) (where Ω = ω on squarefrees). The proof unfolds the definitions and applies moebius_apply_of_squarefree from mathlib. Verified locally against mathlib v4.29.0 (PNT is on v4.30.0; the lemmas used are stable core mathlib API).
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Gentle ping — happy to address any feedback or rebase if needed. The patch is a 4-line proof with no new imports and is verified against mathlib v4.29.0. Thanks for taking a look whenever you have a moment! |
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Unless this is Vaishnav, this task is already claimed. |
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Summary
This PR proves the
sorryinliouville_eq_moebius_on_squarefreeinPrimeNumberTheoremAnd/IwaniecKowalskiCh1.lean(line 1456 onmain).Strategy
For squarefree
n:n ≠ 0(fromnot_squarefree_zero)liouville n = (-1)^Ω(n)by definition (unfoldingtoArithmeticFunction)μ n = (-1)^Ω(n)bymoebius_apply_of_squarefreeBoth sides reduce to the same expression.
Notes
not_squarefree_zero,moebius_apply_of_squarefree— are stable core mathlib API present in both v4.29.0 and v4.30.0).