Large Z(t) values – Riemann Zeta Search Project

Applying the RS-PEAK algorithm the Riemann Zeta Search Project found millions of large values where $Z(t)>1.000$ .

Define the function $\varphi(t)=\frac{\log{|Z(t)|}}{\log(t)}$ . From the theorem of Bourgain $\zeta(1/2+it)= O(t^{13/84+\epsilon})$ for every $\epsilon>0$ , and assuming the Riemann hypothesis $\lim\limits_{t \rightarrow \infty}{\varphi(t)}=0\, .$ Finding extremely large $Z(t)$ values with large $\varphi(t)$ is challenging.

The top 37 largest $Z(t)$ with the appropriate $\varphi(t)$ found and verified by the Riemann Zeta Search Project are the followings:

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