Research proof index

Serre derivative is equivariant under the slash action. More precisely, if F is invariant under the slash action of weight k, then serre_D k F is invariant under the slash action of weight k + 2.

General limit: if f → c at i∞ and f is holomorphic and bounded, then serre_D k f → -k*c/12. This is the continuous mapping theorem applied to serre_D k f = D f - (k/12) * E₂ * f: - D f → 0 (Cauchy estimate from boundedness) - E₂ → 1 - f → c Therefore serre_D k f → 0 - (k/12) * 1 * c = -k*c/12.

The Serre derivative of E₂ is weight-4 slash-invariant. This requires explicit computation since E₂ is not modular. Proof strategy: Write serre_D 1 E₂ = serre_D 2 E₂ + (1/12) E₂². Then: - (serre_D 2 E₂) ∣[4] γ = serre_D 2 (E₂ ∣[2] γ) by serre_D_slash_equivariant - E₂ ∣[2] γ = E₂ - α D₂ γ where α = 1/(2ζ(2)) = 3/π² - (E₂²) ∣[4] γ = (E₂ ∣[2] γ)² After e...

Generic derivative bound for σ_k q-series on compact sets. Uses σ_k(n) ≤ n^(k+1) (sigma_bound) and iter_deriv_comp_bound3 for exponential decay.

Generic summability for n^a * σ_b(n) * exp(2πinz) series. Uses σ_b(n) ≤ n^(b+1) (sigma_bound) and a33 (a+b+1) for exponential summability.

If G is generated by a set s, then the slash action by elements in G is uniquely determined by the slash action by elements in s. See slashaction_generators' for a version where s is a set of elements in G.