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

On RH we have

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


\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 ,

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.

Back to the main index for L-functions and Random Matrix Theory.