Integral Representation for Riemann-Siegel $Z(t)$ function

TL;DR

This paper introduces an integral representation of the Riemann-Siegel Z(t) function using the Poisson formula, providing a new perspective that could aid in understanding the zeros of the zeta function on the critical line.

Contribution

It presents a novel integral formula for Z(t) and derives an asymptotic estimate, connecting the function's behavior to the zeros of the zeta function.

Findings

Derived an integral representation of Z(t) involving the zeta function.

Obtained an asymptotic estimate for Z(t) with an explicit error term.

Linked the study of Z(t) to the distribution of zeta zeros.

Abstract

We apply Poisson formula for a strip to give a representation of $Z(t)$ by means of an integral. \[F(t)=\int_{-\infty}^\infty \frac{h(x)\zeta(4+ix)}{7\cosh\pi\frac{x-t}{7}}\,dx, \qquad Z(t)=\frac{\Re F(t)}{(\frac14+t^2)^{\frac12}(\frac{25}{4}+t^2)^{\frac12}}.\] After that we get the estimate \[Z(t)=\Bigl(\frac{t}{2\pi}\Bigr)^{\frac74}\Re\bigl\{e^{i\vartheta(t)}H(t)\bigr\}+O(t^{-3/4}),\] with \[H(t)=\int_{-\infty}^\infty\Bigl(\frac{t}{2\pi}\Bigr)^{ix/2}\frac{\zeta(4+it+ix)}{7\cosh(\pi x/7)}\,dx=\Bigl(\frac{t}{2\pi}\Bigr)^{-\frac74}\sum_{n=1}^\infty \frac{1}{n^{\frac12+it}}\frac{2}{1+(\frac{t}{2\pi n^2})^{-7/2}}.\] We explain how the study of this function can lead to information about the zeros of the zeta function on the critical line.