Theorems · Theorem · number theory
riemannZeta_eulerProduct
∀ {s : ℂ},
1 < s.re → Filter.Tendsto (fun n => ∏ p ∈ n.primesBelow, (1 - ↑p ^ (-s))⁻¹) Filter.atTop (nhds (riemannZeta s))The Euler product for the Riemann ζ function, valid for s.re > 1.
This version is stated in the form of convergence of finite partial products.
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- Foundations
- Depth 307 from the axioms · uses propext, Classical.choice, Quot.sound
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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
- nhdsstatement and proof · cited by 5,554
- Filter.Tendstostatement and proof · cited by 3,814
- Filter.atTopstatement and proof · cited by 2,405
- Finset.prodstatement and proof · cited by 2,356
- Complex.restatement and proof · cited by 882
- riemannZetastatement · cited by 85
- Nat.primesBelowstatement and proof · cited by 43
- summable_riemannZetaSummandproof · cited by 4
- tsum_riemannZetaSummandproof · cited by 2
- EulerProduct.eulerProduct_completely_multiplicativeproof · cited by 2
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