An expression for Riemann Siegel function

TL;DR

This paper derives an expression for the Riemann-Siegel function Z(t) using the zeros of a specific entire function related to Riemann's work, providing insights into the zeros of the zeta function.

Contribution

It presents a new expression for Z(t) in terms of the zeros of an entire function al(s) connected to Riemann's original formulations.

Findings

Expresses Z(t) via zeros of al(s)

Links al(s) to Riemann's original work on al(s)

Provides a new perspective on zeros of the zeta function

Abstract

There are many analytic functions $U(t)$ satisfying $Z(t)=2\Re\bigl\{ e^{i\vartheta(t)}U(t)\bigr\}$. Here, we consider an entire function $\mathop{\mathcal L}(s)$ such that $U(t)=\mathop{\mathcal L}(\frac12+it)$ is one of the simplest among them. We obtain an expression for the Riemann-Siegel function $Z(t)$ in terms of the zeros of $\mathop{\mathcal L}(s)$. Implicitly, the function $\mathop{\mathcal L}(s)$ is considered by Riemann in his paper on Number Theory. Riemann spoke of having used an expression for $\Xi(t)$ in his demonstration that most of the non-trivial zeros of the zeta function lie on the critical line. Therefore, any expression deserves a study.