Theorems · Theorem · number theory
HurwitzZeta.hasSum_nat_cosZeta
∀ (a : ℝ) {s : ℂ}, 1 < s.re → HasSum (fun n => ↑(Real.cos (2 * Real.pi * a * ↑n)) / ↑n ^ s) (HurwitzZeta.cosZeta (↑a) s)Formula for cosZeta as a Dirichlet series in the convergence range, with sum over ℕ.
- Cited by
- 4 results in Mathlib
- Foundations
- Depth 303 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites34
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- Realstatement and proof · cited by 25,697
- Complexstatement and proof · cited by 5,565
- add_zeroproof · cited by 2,707
- SummationFilter.unconditionalstatement and proof · cited by 2,068
- absproof · cited by 1,814
- Real.pistatement and proof · cited by 1,774
- Complex.ofRealstatement and proof · cited by 1,654
- Complex.restatement and proof · cited by 882
- Complex.Iproof · cited by 866
- neg_mulproof · cited by 654
- Complex.expproof · cited by 612
- mul_negproof · cited by 590
Cited by4
Results whose statement or proof uses this declaration.
- zeta_eq_tsum_one_div_nat_cpowproof · cited by 5
- HurwitzZeta.cosZeta_two_mul_natproof · cited by 1
- HurwitzZeta.hasSum_expZeta_of_one_lt_reproof · cited by 1
- HurwitzZeta.LSeriesHasSum_cosproof · cited by 0