Approximate formula for $Z(t)$

Abstract

The series for the zeta function does not converge on the critical line but the function \[G(t)=\sum_{n=1}^\infty \frac{1}{n^{\frac12+it}}\frac{t}{2πn^2+t}\] satisfies $Z(t)=2\Re \{e^{i\vartheta (t)}G(t)\}+O(t^{-\frac{5}{6}+\epsilon })$. 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 $Z(t)=2\Re \{e^{i\vartheta (t)}U(t)\}$.

arXiv comment12 pages 3 figures