OAI/NumberTheory/Ostmann/ZeroDensity/ImprimitiveEulerBound.lean

GitHub source file

8 declarations, 0 contain sorry, 0 declared as axioms

The finite Euler-factor cost of changing a character's level

  1. def OAI.Ostmann.imprimitiveEulerFactor line 13
  2. def OAI.Ostmann.imprimitiveEulerProduct line 16
  3. theorem OAI.Ostmann.euler_tail_norm_le_half line 19
  4. theorem OAI.Ostmann.imprimitiveEulerFactor_ne_zero line 31
  5. theorem OAI.Ostmann.imprimitiveEulerFactor_hasDeriv line 39
  6. theorem OAI.Ostmann.imprimitiveEulerFactor_logDeriv_bound line 49
  7. theorem OAI.Ostmann.imprimitiveEulerProduct_logDeriv_bound line 69
  8. theorem OAI.Ostmann.logDeriv_changeLevel_bound line 89

    Source documentation

    Passing from a primitive character to any multiple level costs at most log N in its logarithmic derivative, uniformly throughout Re(s)>1.

    theorem logDeriv_changeLevel_bound {q N : ℕ} [NeZero q] [NeZero N]
    (χ : DirichletCharacter ℂ q) (hN : q ∣ N) (s : ℂ) (hs : 1 < s.re) :
    ‖logDeriv (DirichletCharacter.LFunction (DirichletCharacter.changeLevel hN χ)) s -
      logDeriv (DirichletCharacter.LFunction χ) s‖ ≤ Real.log N := by
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