The Laplacian of The Integral Of The Logarithmic Derivative of the Riemann-Siegel-Hardy Z-function

Abstract

The integral $R(t)={\pi }^{-1}(ln\zeta (\frac{1}{2}+it)+i\vartheta (t))$ of the logarithmic derivative of the Hardy Z function $Z(t)=e^{i\vartheta (t)}\zeta (\frac{1}{2}+it)$, where $\vartheta (t)$ is the Riemann-Siegel theta function, and $\zeta (t)$ is the Riemann zeta function, is used as a basis for the construction of a pair of transcendental entire functions $\nu (t)=-\nu (1-t)=-{\Delta R(\frac{i}{2}-it)}^{-1}=-G(\frac{i}{2}-it)$ where $G=-(\Delta R(t){)}^{-1}$ is the derivative of the additive inverse of the reciprocal of the Laplacian of $R(t)$ and $\chi (t)=-\chi (1-t)=\dot{\nu }(t)=-iH(\frac{i}{2}-it)$ where $H(t)=\dot{G}(t)$ has roots at the local minima and maxima of $G(t)$. When $H(t)=0$ and $\dot{H}(t)=\ddot{G}(t)=\Delta G(t)>0$, the point $t$ marks a minimum of $G(t)$ where it coincides with a Riemann zero, i.e., $\zeta (\frac{1}{2}+it)=0$, otherwise when $H(t)=0$ and $\dot{H}(t)=\Delta G(t)<0$, the point $t$ marks a local maximum of $G(t)$, marking midway points between consecutive minima. Considered as a sequence of distributions or wave functions, ${\nu }_n(t)=\nu (1+2n+2t)$ converges to ${\nu }_{\infty }(t)=li{m}_{n\to \infty }{\nu }_n(t)={\sin }^2(\pi t)$ and ${\chi }_n(t)=\chi (1+2n+2t)$ to ${\chi }_{\infty }(t)=li{m}_{n\to \infty }{\chi }_n(t)=-8\cos (\pi t)\sin (\pi t)$

References in corpus (1)

• Physics of the Riemann Hypothesis