Both the constant $\pi$ and the constant $e$ have clear algebraic roles in complex numbers and in differential calculus.

But did anyone ever propose an algebraic role for Euler-Mascheroni constant $\gamma$?

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    $\begingroup$ What is "algebraic role"? Just some analytic identities involving it? $\endgroup$ – Conifold Apr 8 at 21:00
  • $\begingroup$ @Conifold well, algebraic role is serving as unity, invariant or fixed point under certain basic operations or being a result of basic operations applied to unity or zero. For instance, $e^x$ is invariant under differentiation, and $e^{1/e}$ is fixed point of tetration. $\endgroup$ – Anixx Apr 8 at 21:17
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    $\begingroup$ Well, $-\gamma=\psi(1)$ like $e=\exp(1)$, but the digamma $\psi$ is more analogous to $\ln$ than to $\exp$. It has the same relation to the difference operator, $\psi(x+1)-\psi(x)=\frac1x$, that $\ln x$ has to the derivative, $(\ln x)'=\frac1x$. So what would be the algebraic role for something like $\ln 2$? $\endgroup$ – Conifold Apr 10 at 8:41

The answer is yes: J. Lagarias, Euler's constant: Euler's work and modern developments, BAMS, 50 (2013), no. 4, 527–628.

  • $\begingroup$ The paper in behind paywall... Can you please outline the proposal briefly? $\endgroup$ – Anixx Apr 8 at 23:49
  • $\begingroup$ I see prompt for password. But it seems, the paper is published in Arxiv arxiv.org/abs/1303.1856 $\endgroup$ – Anixx Apr 8 at 23:55
  • $\begingroup$ I changed the link. Please try again. $\endgroup$ – Alexandre Eremenko Apr 9 at 0:04
  • $\begingroup$ Yes, now the link works. But the paper is quite big and makes an overview. Did you mean some particular statement in the paper as the one which highlights the algebraic role? $\endgroup$ – Anixx Apr 9 at 0:07
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    $\begingroup$ @Anixx: No, I mean that the whole paper answers your question. It surveys the role that Euler's constant plays in a variety of questions. There is no point in condensing it in few lines. If you are really interested, you will enjoy it. And I also do not understand what "algebraic role" means. $\endgroup$ – Alexandre Eremenko Apr 9 at 0:24

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