I performed some computations in wolfram alpha looking at the behavior of the values of $|\zeta(e^{ni})|$ trying to predict a lower bound. I have got the following result:

For $n > 19 :|\zeta(e^{ni})|\leq \log(n)$

My question here is: Is the above result true and if it is, how can I show it?

Note :$i$ is the unit imaginary part of complex number and $n$ is a positive integer

Thank you for any help

  • 6
    $\begingroup$ I take it $\zeta$ is the Riemann zeta-function. It has a pole at $z=1$, so if $n$ is close to a multiple of $2\pi$ you'll get large values. This will happen if $n$ is the numerator of a continued fraction approximant to $2\pi$, so that's where I'd look for counterexamples. $\endgroup$ – Gerry Myerson Oct 20 '16 at 22:03
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    $\begingroup$ Indeed, per @GerryMyerson 's suggestion, $n=710$ is a pretty convincing counterexample. $\endgroup$ – Steven Landsburg Oct 20 '16 at 22:30
  • $\begingroup$ I rolled this back because the recent edits were screwing up the LaTeX. $\endgroup$ – Steven Landsburg Oct 20 '16 at 23:58

The minimal counterexample is $n=25$ (see the numerical verification).

Another small counterexample is $n=44$, which as suggested by Gerry Myerson comes from the approximation $2\pi \approx 44/7$.


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