An expression for Riemann Siegel 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.