  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 $Z(t)=2\Re\{e^{i\vartheta(t)}U(t)\}$.
