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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2$\begingroup$ What is "algebraic role"? Just some analytic identities involving it? $\endgroup$– ConifoldCommented Apr 8, 2021 at 21:00
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$\begingroup$ @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. $\endgroup$– AnixxCommented Apr 8, 2021 at 21:17
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1$\begingroup$ 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 $\ln 2$? $\endgroup$– ConifoldCommented Apr 10, 2021 at 8:41
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The answer is yes: J. Lagarias, Euler's constant: Euler's work and modern developments, BAMS, 50 (2013), no. 4, 527–628.
answered Apr 8, 2021 at 23:48
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$\begingroup$ The paper in behind paywall... Can you please outline the proposal briefly? $\endgroup$– AnixxCommented Apr 8, 2021 at 23:49
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$\begingroup$ I see prompt for password. But it seems, the paper is published in Arxiv arxiv.org/abs/1303.1856 $\endgroup$– AnixxCommented Apr 8, 2021 at 23:55
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$\begingroup$ I changed the link. Please try again. $\endgroup$ Commented Apr 9, 2021 at 0:04
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$\begingroup$ 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? $\endgroup$– AnixxCommented Apr 9, 2021 at 0:07
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3$\begingroup$ @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. $\endgroup$ Commented Apr 9, 2021 at 0:24