OAI/NumberTheory/Ostmann/ZeroDensity/RealCompletedHadamard.lean

GitHub source file

5 declarations, 0 contain sorry, 0 declared as axioms

The real-axis Hadamard identity from actual disk sums

  1. def OAI.Ostmann.realCharacterZeroSum line 15
  2. theorem OAI.Ostmann.completed_partial_real_limit line 19
  3. theorem OAI.Ostmann.real_zero_kernel_reflection line 35
  4. theorem OAI.Ostmann.real_zero_sum_reflection line 46
  5. theorem OAI.Ostmann.real_completed_hadamard_identity line 64

    Source documentation

    Reflection cancels the constant term in the canonical product.

    theorem real_completed_hadamard_identity (χ : PrimitiveRealCharacter) (s : ℝ)
    (hs : 1 < s) (hs2 : s ≤ 2) :
    realCharacterZeroSum χ s = (logDeriv χ.asComplex.completed (s : ℂ)).re +
      (1 / 2) * Real.log χ.modulus := by
…

    Mentions: OAI.Ostmann.PrimitiveRealCharacter, OAI.Ostmann.realCharacterZeroSum