Theorems · Theorem · number theory
HurwitzZeta.cosZeta_two_mul_nat
∀ {k : ℕ} {x : ℝ},
k ≠ 0 →
x ∈ Set.Icc 0 1 →
HurwitzZeta.cosZeta (↑x) (2 * ↑k) =
(-1) ^ (k + 1) * (2 * ↑Real.pi) ^ (2 * k) / 2 / ↑(2 * k).factorial *
Polynomial.eval (↑x) (Polynomial.map (algebraMap ℚ ℂ) (Polynomial.bernoulli (2 * k)))Express the value of cosZeta at a positive even integer as a value
of the Bernoulli polynomial.
- Cited by
- 1 results in Mathlib
- Foundations
- Depth 304 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites40
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- DFunLike.coeproof · cited by 62,936
- Setstatement · cited by 53,352
- Realstatement and proof · cited by 25,697
- Polynomialproof · cited by 5,681
- Complexstatement and proof · cited by 5,565
- Algebra.algebraMapstatement and proof · cited by 4,706
- mul_oneproof · cited by 3,885
- add_zeroproof · cited by 2,707
- Nat.cast_oneproof · cited by 2,501
- SummationFilter.unconditionalproof · cited by 2,068
- Real.pistatement and proof · cited by 1,774
- Set.Iccstatement and proof · cited by 1,702
Cited by1
Results whose statement or proof uses this declaration.
- HurwitzZeta.cosZeta_two_mul_nat'proof · cited by 1