Theorems · Theorem · number theory
zeta_eq_tsum_one_div_nat_cpow
∀ {s : ℂ}, 1 < s.re → riemannZeta s = ∑' (n : ℕ), 1 / ↑n ^ sThe Riemann zeta function agrees with the naive Dirichlet-series definition when the latter
converges. (Note that this is false without the assumption: when re s ≤ 1 the sum is divergent,
and we use a different definition to obtain the analytic continuation to all s.)
- Defined in
- Mathlib.NumberTheory.LSeries.RiemannZeta
- Cited by
- 5 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.
Cites15
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- Realstatement · cited by 25,697
- Complexstatement and proof · cited by 5,565
- MulZeroClass.mul_zeroproof · cited by 2,091
- SummationFilter.unconditionalstatement and proof · cited by 2,068
- Real.piproof · cited by 1,774
- Complex.ofRealproof · cited by 1,654
- MulZeroClass.zero_mulproof · cited by 1,625
- tsumstatement and proof · cited by 1,148
- Complex.restatement and proof · cited by 882
- Real.cosproof · cited by 424
- HasSum.tsum_eqproof · cited by 150
- riemannZetastatement · cited by 85
Cited by5
Results whose statement or proof uses this declaration.
- zeta_nat_eq_tsum_of_gt_oneproof · cited by 4
- zeta_eq_tsum_one_div_nat_add_one_cpowproof · cited by 3
- ArithmeticFunction.LSeries_zeta_eq_riemannZetaproof · cited by 2
- tendsto_sub_mul_tsum_nat_cpowproof · cited by 1
- riemannZeta_conjproof · cited by 0