OAI/NumberTheory/Ostmann/Preliminaries/MertensThetaKernel.lean

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4 declarations, 0 contain sorry, 0 declared as axioms

Integrating the conductor-one theta error

The exponential square-root error has a bounded logarithmic integral. This concrete antiderivative suffices for the first Mertens estimate.

  1. def OAI.Ostmann.thetaErrorPrimitive line 17
  2. theorem OAI.Ostmann.hasDerivAt_thetaErrorPrimitive line 21
  3. theorem OAI.Ostmann.thetaErrorPrimitive_nonpos line 36
  4. theorem OAI.Ostmann.theta_error_kernel_integral_bound line 44

    Source documentation

    A uniform bound independent of the upper cutoff.

    theorem theta_error_kernel_integral_bound {c X : ℝ} (hc : 0 < c) (hX : 2 ≤ X) :
    (∫ t in (2 : ℝ)..X, Real.exp (-c * Real.sqrt (Real.log t)) / t) ≤
      -thetaErrorPrimitive c 2 := by
…

    Mentions: OAI.Ostmann.thetaErrorPrimitive