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

TL;DR

This paper introduces new transcendental functions derived from the Riemann zeta function's integral and explores their properties, roots, and limits, aiming to shed light on the distribution of Riemann zeros.

Contribution

It constructs novel entire functions related to the Riemann zeta function and analyzes their roots and asymptotic behavior, providing new insights into the zeros of the zeta function.

Findings

Functions $ u(t)$ and $ ext{ extgreek heta}(t)$ have roots at Riemann zeros.

Sequences of these functions converge to sine and cosine functions.

The analysis links the extrema of these functions to the distribution of zeros.

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 $\chi_\infty (t)=lim_{n \rightarrow \infty} \chi_n (t)=-8 \cos(\pi t) \sin(\pi t)$