Approximate formula for $Z(t)$

TL;DR

This paper introduces an approximate formula for the Riemann zeta function on the critical line, linking it to a new series representation that closely predicts the zeros of the zeta function.

Contribution

The paper proposes a novel series-based approximation for the zeta function, providing insights into the location of its zeros on the critical line.

Findings

The series G(t) approximates Z(t) with an error of O(t^{-5/6+ε}).

Zeros of Z(t) are near zeros of the real part of e^{iθ(t)}G(t).

A related function U(t) also satisfies a similar relation with Z(t).

Abstract

The series for the zeta function does not converge on the critical line but the function \[G(t)=\sum_{n=1}^\infty \frac{1}{n^{\frac12+it}}\frac{t}{2\pi n^2+t}\] satisfies $Z(t)=2\Re\{e^{i\vartheta(t)}G(t)\}+O(t^{-\frac56+\varepsilon})$. So one expects that the zeros of zeta on the critical line are very near the zeros of $\Re\{e^{i\vartheta(t)}G(t)\}$. There is a related function $U(t)$ that satisfies the equality $Z(t)=2\Re\{e^{i\vartheta(t)}U(t)\}$.