Research proof index

For a cusp form f of level Γ(n), we have t^s * f(it) → 0 as t → ∞ for any real power s. This follows from the exponential decay of cusp forms at infinity: f = O(exp(-2π τ.im / n)).

If F has a Fourier expansion ∑_{m≥0} a_m exp(2πi(m+n₀)z) with n₀ > 0, and the coefficients are absolutely summable at height im z = c, then F = O(exp(-2π n₀ · im z)) at atImInfty. The key bound is: for im z ≥ c, ‖F(z)‖ ≤ (∑_m ‖a_m‖ · exp(-2π c m)) · exp(-2π n₀ · im z)

If F : ℍ → ℂ is O(exp(-c * im τ)) at atImInfty for some c > 0, then the restriction to the imaginary axis t ↦ F(it) is O(exp(-c * t)) at atTop.

For any function F : ℍ → ℂ and t > 0, F.resToImagAxis (1/t) = F(S • (I*t)).

Real part of a product on the imaginary axis is the product of real parts, when both factors are real-valued there.

Restriction and slash action under S: $(F |_k S) (it) = (it)^{-k} * F(it)$