Theorems · Theorem · number theory
DirichletCharacter.LFunction_ne_zero_of_one_le_re
∀ {N : ℕ} (χ : DirichletCharacter ℂ N) [inst : NeZero N] ⦃s : ℂ⦄,
χ ≠ 1 ∨ s ≠ 1 → 1 ≤ s.re → DirichletCharacter.LFunction χ s ≠ 0If χ is a Dirichlet character, then L(χ, s) does not vanish for s.re ≥ 1
except when χ is trivial and s = 1 (then L(χ, s) has a simple pole at s = 1).
- Cited by
- 5 results in Mathlib
- Foundations
- Depth 317 from the axioms · uses propext, Classical.choice, Quot.sound
- Assumes
- NeZero
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites10
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
- ZModstatement · cited by 1,024
- Complex.restatement and proof · cited by 882
- LE.le.eq_or_ltproof · cited by 220
- DirichletCharacterstatement and proof · cited by 161
- DirichletCharacter.LFunctionstatement · cited by 23
- DirichletCharacter.LFunction_eq_LSeriesproof · cited by 4
- DirichletCharacter.LSeries_ne_zero_of_one_lt_reproof · cited by 2
- DirichletCharacter.LFunction_ne_zero_of_re_eq_oneproof · cited by 1
Cited by5
Results whose statement or proof uses this declaration.
- ArithmeticFunction.vonMangoldt.eqOn_LFunctionResidueClassAuxproof · cited by 2
- ArithmeticFunction.vonMangoldt.continuousOn_LFunctionResidueClassAuxproof · cited by 1
- ArithmeticFunction.vonMangoldt.continuousOn_LFunctionResidueClassAux'proof · cited by 1
- DirichletCharacter.LFunction_apply_one_ne_zeroproof · cited by 0
- riemannZeta_ne_zero_of_one_le_reproof · cited by 0