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

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 $\nu (t)=-\nu (1-t)=-{\mathrm{\Delta }R(\frac{i}{2}-it)}^{-1}=-G(\frac{i}{2}-it)$ where $G=-(\mathrm{\Delta }R(t){)}^{-1}$ 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 $\dot{H} (t)=\ddot{G} (t)={\Delta}G(t)>0$,thepoint$t$marksaminimumof$G(t)$ where it coincides with a Riemann zero, i.e., $\zeta (\frac{1}{2}+it)=0$, otherwise when $H(t)=0$ and $\dot{H}(t)=\mathrm{\Delta }G(t)<0$, 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 ${\nu }_{\mathrm{\infty }}(t)=li{m}_{n\to \mathrm{\infty }}{\nu }_n(t)={\sin }^{}(\pi t)$ and ${\chi }_n(t)=\chi (1+2n+2t)$ to $\chi_\infty (t)=lim_{n \rightarrow \infty} \chi_n (t)=-8 \cos(\pi t) \sin(\pi t)$