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Euler’s constant: Euler’s work and modern developments. (English) Zbl 1282.11002
This article, which deals with Euler’s work related to the Euler constant \(\gamma\) and modern developments, is actually a small book. On 100 pages, the author gives an excellent survey on the mathematics of Euler’s constant, which is defined as the limit \[ \lim_{n \to \infty} \Big( \sum_{j=1}^n \frac1n - \log n \Big). \] It begins with two conjectures, namely that a) \(\gamma\) is irrational and that b) it is not a period in the sense of Kontsevich and Zagier. The first quarter of this article presents Euler’s work on \(\gamma\), with detailed references. The remaining three quarters deal with the subsequent developments related to the Gamma function, the Riemann zeta function, other arithmetic functions, random permutations, periods, diophantine approximations and transcendence results. This is a must-read for every mathematician and for everyone interested in Euler’s work.

MSC:
11-02 Research exposition (monographs, survey articles) pertaining to number theory
11-03 History of number theory
01A50 History of mathematics in the 18th century
11Y60 Evaluation of number-theoretic constants
11J72 Irrationality; linear independence over a field
11J81 Transcendence (general theory)
11M06 \(\zeta (s)\) and \(L(s, \chi)\)
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