Euler's constant: Euler's work and modern developments

Abstract

This paper has two parts. The first part surveys Euler's work on the constant gamma=0.57721... bearing his name, together with some of his related work on the gamma function, values of the zeta function and divergent series. The second part describes various mathematical developments involving Euler's constant, as well as another constant, the Euler-Gompertz constant. These developments include connections with arithmetic functions and the Riemann hypothesis, and with sieve methods, random permutations and random matrix products. It includes recent results on Diophantine approximation and transcendence related to Euler's constant.

v4 98 pages (pagewidth decreased), 321 references, text agrees with published version (but with less reference information); v5 post-publication, introduces a mistake, Theorem 3.1.3 is ok as originally stated, v6, like the monkey's paw, changes back, but corrects typo in (3.1.8) and in ζ(1-k) formula two lines following

References in corpus (4)

• De seriebus divergentibus
• The generalized-Euler-constant function $\gamma (z)$ and a generalization of Somos's quadratic recurrence constant
• On linear forms containing the Euler constant
• Dickman polylogarithms and their constants

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• Numerical verification of Littlewood's bounds for $|L(1,\chi )|$
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• Notes on the historical bibliography of the gamma function
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• Linear and algebraic independence of Generalized Euler-Briggs constants
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• Basel problem: a physicist's solution
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