The order of the $\zeta$-function on the 1-line

On RH we have

\begin{displaymath} e^\gamma \le \limsup_{t\to\infty} \frac{\zeta(1+it)}{\log\log t} \le 2 e^\gamma , \end{displaymath}

and

\begin{displaymath} \frac{6}{\pi^2} e^\gamma \le \limsup_{t\to\infty} \frac{1/\zeta(1+it)}{\log\log t} \le \frac{12}{\pi^2} e^\gamma , \end{displaymath}

where $\gamma$ is Euler's constant. See Titchmarsh [ MR 88c:11049] for proofs.

Since these estimates concern non-critical values of an $L$-function, one might suspect that the smaller of each of the above results is the true answer.



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