An expression for Riemann Siegel function

Abstract

There are many analytic functions $U(t)$ satisfying $Z(t)=2\Re \{e^{i\vartheta (t)}U(t)\}$. Here, we consider an entire function $\mathcal{L}(s)$ such that $U(t)=\mathcal{L}(\frac{1}{2}+it)$ is one of the simplest among them. We obtain an expression for the Riemann-Siegel function $Z(t)$ in terms of the zeros of $\mathcal{L}(s)$. Implicitly, the function $\mathcal{L}(s)$ 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.

arXiv comment10 pages 2 figures