2000 character limit reached
Approximate formula for $Z(t)$ (2406.18150v1)
Published 26 Jun 2024 in math.NT
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 n2+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 $Z(t)=2\Re{e{i\vartheta(t)}U(t)}$.