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 $Z(t)=2\Re\{e^{i\vartheta(t)}G(t)\}+O(t^{-\frac56+\varepsilon})$. 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 $Z(t)=2\Re\{e^{i\vartheta(t)}U(t)\}$.

Figures (4)

• Figure 1: Plot of $G(t)$ in $(1000,1040)\times(-10,10)$.
• Figure 2: x-ray of $e^{i\vartheta(t)}G(t)$ in $(200\,040,200\,060)\times(-2,4)$
• Figure 3: Plot of $G(t)$ in $(50,100)\times(-20,20)$.
• Figure 4: Plot of $G(t)$ in $(1000,1040)\times(-10,10)$.

Theorems & Definitions (21)

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