English
For any Dirichlet character χ, the twisted L-series identity holds: (χ * Λ) ⍟ χ = χ * log, i.e., the χ-twisted convolution equals χ times the logarithm function.
Русский
Для любого персонажа Дирichле выполняется твиcт-отношение: (χ * Λ) ⍟ χ = χ * log, то есть χ–свёртка равна χ умноженному на логарифм.
LaTeX
$$$(\\uparrow χ * \\uparrow Λ) ⍟ \\uparrow χ = \\uparrow χ * \\uparrow \\Complex.log$$$
Lean4
/-- A twisted version of the relation `Λ * ↑ζ = log` in terms of complex sequences. -/
theorem convolution_twist_vonMangoldt {N : ℕ} (χ : DirichletCharacter ℂ N) : (↗χ * ↗Λ) ⍟ ↗χ = ↗χ * ↗Complex.log := by
rw [← convolution_vonMangoldt_const_one, ← χ.mul_convolution_distrib, mul_one]