Let $\gamma$ be the Euler-Mascheroni constant. Why is $\gamma$ not a Liouville number? Are there any upper bounds for the irrationality measure of $\gamma$ known?

Any pointers to the literature are welcome. I don't find much on this topic online. Thanks.

  • $\begingroup$ Many terms of the continued fraction of $\gamma$ has been computed. oeis.org/A002852 it doesn't look like it's particularly well-approximated by rationals so it's almost certainly not a Liouville number. $\endgroup$ Apr 30, 2017 at 23:52

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At the present time, we do not even know how to prove that the Euler-Mascheroni constant $\gamma=\lim_{n\to\infty} \sum_{k=1}^n\frac{1}{k} - \log n$ is irrational, much less transcendental; although it is conjectured to be transcendental. The reason you won't find a lot about this topic online (or in the research literature) is because so little is known about the algebraic properties of $\gamma$.

There is an analgoue of the Euler-Mascheroni constant for Carlitz modules, and I know that there are many transcendence results in the Carlitz (and Drin'feld and T)-module universe that aren't currently provable over number fields. See for example

Tensor Powers of the Carlitz Module and Zeta Values, Greg W. Anderson; Dinesh S. Thakur, The Annals of Mathematics, 2nd Ser., Vol. 132, No. 1. (Jul., 1990), pp. 159-191.

But I don't know if irrationality (or transcendence) has even been proven in that setting.

  • $\begingroup$ Thanks, Prof. Silverman. I will look at the Carlitz modules. Even if it is unknown that $\gamma$ is irrational, one might establish that $\gamma$ is not a Liouville number independently. $\endgroup$
    – Christoph
    May 1, 2017 at 9:36

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