Research proof index

G(1/s) = s^10 * (H₄(is))³ * (2(H₄(is))² + 5H₂(is)H₄(is) + 5(H₂(is))²)

Functional equation of $G$ restricted to the imaginary axis.

G / q^(3/2) → 20480 as im(z) → ∞. Here q^(3/2) = exp(2πi · (3/2) · z).

A weighted Hölder lower bound for Fourier energy. If $\Delta$ lies in the $\eta$-large spectrum of $f$, $m\ne0$, and a weight $\nu$ is at least $1$ wherever $f$ is nonzero, then the order-$m$ energy of $\Delta$ weighted by $\widehat\nu$ is at least $|\Delta|^{2m}\eta^{2m}\|f\|_1^2/\|f\|_2^2$.

For every real $x\ge2$, the extended-nonnegative geometric series satisfies

$\sum_{n=0}^{\infty}2^{-n/x}\le2^x.$

If one grid cube $I$ is strictly contained below another grid cube $J$, then the project’s phase distance at the finer cube is controlled by the phase distance at the coarser cube:

$d_I(f,g)\le C(a)\,d_J(f,g).$