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 $\nu_\infty (t)=lim_{n \rightarrow \infty} \nu_n (t)={\sin}^2(\pi t)$ and $\chi_n(t)=\chi(1+2n+2t)$ to $\chi_\infty (t)=lim_{n \rightarrow \infty} \chi_n (t)=-8 \cos(\pi t) \sin(\pi t)$