OAI/NumberTheory/Ostmann/ZeroDensity/ShiftedRealHadamard.lean

GitHub source file

7 declarations, 0 contain sorry, 0 declared as axioms

Real parts of the Hadamard identity off the real axis

  1. theorem OAI.Ostmann.realZeroKernel_complex_sub line 12
  2. theorem OAI.Ostmann.inverse_square_shift_bound line 17
  3. def OAI.Ostmann.shiftedRealCharacterZeroSum line 35
  4. theorem OAI.Ostmann.shifted_real_zero_summable line 38
  5. theorem OAI.Ostmann.shifted_completed_partial_real_limit line 57
  6. theorem OAI.Ostmann.shifted_real_zero_reflection line 72
  7. theorem OAI.Ostmann.shifted_real_hadamard_identity line 95

    Source documentation

    The shifted identity supplies the actual zero terms used in the low-height zero-free-region argument.

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

    Mentions: OAI.Ostmann.PrimitiveRealCharacter, OAI.Ostmann.shiftedRealCharacterZeroSum