Theorems · Theorem · number theory
DirichletCharacter.sum_characters_eq
∀ (R : Type u_1) [inst : CommRing R] {n : ℕ} [NeZero n] [HasEnoughRootsOfUnity R (Monoid.exponent (ZMod n)ˣ)]
[inst_3 : IsDomain R] (a : ZMod n), ∑ χ, χ a = if a = 1 then ↑n.totient else 0If R is an integral domain that has enough roots of unity and n ≠ 0, then
for a in ZMod n, the sum of χ a over all Dirichlet characters mod n
with values in R vanishes if a ≠ 1 and has the value n.totient if a = 1.
- Cited by
- 1 results in Mathlib
- Foundations
- Depth 169 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
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Cites19
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
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- Finset.univstatement and proof · cited by 3,473
- Unitsstatement and proof · cited by 2,804
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- IsDomainstatement and proof · cited by 2,196
- ZModstatement and proof · cited by 1,024
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- nsmul_eq_mulproof · cited by 369
Cited by1
Results whose statement or proof uses this declaration.
- DirichletCharacter.sum_char_inv_mul_char_eqproof · cited by 1