Theorems · Theorem · number theory
ArithmeticFunction.vonMangoldt.residueClass_apply
∀ {q : ℕ} {a : ZMod q} [NeZero q],
IsUnit a →
∀ (n : ℕ),
↑(ArithmeticFunction.vonMangoldt.residueClass a n) =
(↑q.totient)⁻¹ * ∑ χ, χ a⁻¹ * χ ↑n * ↑(ArithmeticFunction.vonMangoldt n)We can express ArithmeticFunction.vonMangoldt.residueClass as a linear combination
of twists of the von Mangoldt function by Dirichlet characters.
- Defined in
- Mathlib.NumberTheory.LSeries.PrimesInAP
- Cited by
- 2 results in Mathlib
- Foundations
- Depth 296 from the axioms · uses propext, Classical.choice, Quot.sound
- Assumes
- NeZero
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites23
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- DFunLike.coestatement and proof · cited by 62,936
- Realstatement · cited by 25,697
- Set.ofPredproof · cited by 6,101
- Complexstatement and proof · cited by 5,565
- Finset.sumstatement · cited by 5,195
- Finset.univstatement and proof · cited by 3,473
- MulZeroClass.mul_zeroproof · cited by 2,091
- Nat.cast_zeroproof · cited by 1,870
- Complex.ofRealstatement and proof · cited by 1,654
- MulZeroClass.zero_mulproof · cited by 1,625
- IsUnitstatement and proof · cited by 1,602
- LT.lt.ne'proof · cited by 1,417
Cited by2
Results whose statement or proof uses this declaration.
- ArithmeticFunction.vonMangoldt.LSeries_residueClass_eqproof · cited by 1
- ArithmeticFunction.vonMangoldt.residueClass_eqproof · cited by 0