OAI/NumberTheory/Ostmann/ZeroDensity/RealCharacterTrigonometric.lean

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4 declarations, 0 contain sorry, 0 declared as axioms

The positive trigonometric polynomial for real characters

  1. theorem OAI.Ostmann.real_character_unit_polynomial line 12
  2. theorem OAI.Ostmann.LSeries_term_vertical line 24
  3. theorem OAI.Ostmann.real_character_trigonometric_term_nonneg line 35
  4. theorem OAI.Ostmann.real_character_trigonometric_sum_nonneg line 69

    Source documentation

    The Euler series inherits positivity term by term; no prime distribution input or zero-free assumption is used.

    theorem real_character_trigonometric_sum_nonneg (χ : PrimitiveRealCharacter)
    (σ t : ℝ) (hσ : 1 < σ) :
    0 ≤ 3 * (LSeries (fun n => (ArithmeticFunction.vonMangoldt n : ℂ)) (σ : ℂ)).re +
      4 * (LSeries (fun n => χ.complexCharacter n * (ArithmeticFunction.vonMangoldt n : ℂ))
        ((σ : ℂ) + (t : ℂ) * I)).re +
      (LSeries (fun n => (ArithmeticFunction.vonMangoldt n : ℂ))
        ((σ : ℂ) + ((2 * t : ℝ) : ℂ) * I)).re := by
…

    Mentions: OAI.Ostmann.PrimitiveRealCharacter