OAI/NumberTheory/Ostmann/ZeroDensity/CharacterHadamardBounds.lean

GitHub source file

4 declarations, 0 contain sorry, 0 declared as axioms

Logarithmic bounds from the actual character zero sum

  1. theorem OAI.Ostmann.character_real_zero_sum_nonneg line 12
  2. theorem OAI.Ostmann.character_real_zero_term_le line 19
  3. theorem OAI.Ostmann.exists_character_zero_sum_log_bound line 26
  4. theorem OAI.Ostmann.exists_character_logDeriv_real_lower line 40

    Source documentation

    Positivity of every actual zero contribution gives the uniform lower bound for the ordinary logarithmic derivative used in the zero-free argument.

    theorem exists_character_logDeriv_real_lower : ∃ B : ℝ, 0 < B ∧
    ∀ (χ : PrimitiveComplexCharacter) (s : ℂ), 1 < s.re → s.re ≤ 2 →
      -(1 / 2) * Real.log χ.modulus - B * Real.log (|s.im| + 2) ≤
        (logDeriv χ.L s).re := by
…

    Mentions: OAI.Ostmann.PrimitiveComplexCharacter