Research proof index

H₂(it) > 0 for all t > 0. Blueprint: Lemma 6.43 - H₂ is positive on the imaginary axis. Proof strategy: Each term exp(-π(n+1/2)²t) > 0, so Θ₂(it) > 0, hence H₂ = Θ₂^4 > 0.

Use jacobiTheta₂_functional_equation

These three transformation laws follow directly from tsum definition.

H₄(it) > 0 for all t > 0. Blueprint: Corollary 6.43 - H₄ is positive on the imaginary axis. Proof strategy: Use the modular S-transformation relating H₄ and H₂. From H₄_S_action: (H₄ ∣[2] S) = -H₂ From ResToImagAxis.SlashActionS: relates values at t and 1/t. This gives H₂(i/t) = t² * H₄(it), so H₄(it) > 0 follows from H₂(i/t) > 0.

For $0<y\le1$ and $y/2\le y'\le1$, with both $y$ and $y'$ nonnegative, the negative-side Hilbert kernel satisfies the quantitative regularity estimate

$\lVert k(-y)-k(-y')\rVert \le 2^6\,\frac{1}{|y|}\,\frac{|y-y'|}{|y|}.$

If $\varphi$ is supported in the ball $B(z,R)$, then the oscillatory integral with phase difference $f-g$ satisfies

$\left\lVert\int e^{i(f(x)-g(x))}\varphi(x)\,dx\right\rVert \le C(a)\,\mu(B(z,R))\,\lVert\varphi\rVert_{\mathrm{Hol},\tau;B(z,2R)}\,(1+d_{z,R}(f,g))^{-1/(2a^2+a^3)}.$