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

Abstract

The integral $R(t)=π^{-1}(lnζ(\frac{1}{2}+it)+i\vartheta (t))$ of the logarithmic derivative of the Hardy Z function $Z(t)=e^{i\vartheta (t)}ζ(\frac{1}{2}+it)$, where $\vartheta (t)$ is the Riemann-Siegel theta function, and $ζ(t)$ is the Riemann zeta function, is used as a basis for the construction of a pair of transcendental entire functions $ν(t)=-ν(1-t)=-{ΔR(\frac{i}{2}-it)}^{-1}=-G(\frac{i}{2}-it)$ where $G=-(ΔR(t))^{-1}$ is the derivative of the additive inverse of the reciprocal of the Laplacian of $R(t)$ and $χ(t)=-χ(1-t)=\dotν (t)=-iH(\frac{i}{2}-it)$ where $H(t)=\dot{G} (t)$ has roots at the local minima and maxima of $G(t)$. When $H(t)=0$ and $\dot{H} (t)=\ddot{G} (t)=ΔG(t)>0$, the point $t$ marks a minimum of $G(t)$ where it coincides with a Riemann zero, i.e., $ζ(\frac{1}{2}+it)=0$, otherwise when $H(t)=0$ and $\dot{H} (t)=ΔG(t)<0$, 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, $ν_n(t)=ν(1+2n+2t)$ converges to $ν_\infty (t)=lim_{n \rightarrow \infty} ν_n (t)={\sin}^2(πt)$ and $χ_n(t)=χ(1+2n+2t)$ to $χ_\infty (t)=lim_{n \rightarrow \infty} χ_n (t)=-8 \cos(πt) \sin(πt)$