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


Author: Jeffrey C. Lagarias
Journal: Bull. Amer. Math. Soc. 50 (2013), 527-628
MSC (2010): Primary 11J02; Secondary 01A50, 11J72, 11J81, 11M06
DOI: https://doi.org/10.1090/S0273-0979-2013-01423-X
Published electronically: July 19, 2013
MathSciNet review: 3090422
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Abstract: This paper has two parts. The first part surveys Euler's work on the constant $ \gamma =0.57721\cdots $ 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 also includes recent results on Diophantine approximation and transcendence related to Euler's constant.


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Additional Information

Jeffrey C. Lagarias
Affiliation: Department of Mathematics, University of Michigan, Ann Arbor, Michigan 48109-1043
Email: lagarias@umich.edu

DOI: https://doi.org/10.1090/S0273-0979-2013-01423-X
Received by editor(s): July 1, 2010
Received by editor(s) in revised form: December 24, 2012
Published electronically: July 19, 2013
Additional Notes: The research of the author was supported by NSF Grants DMS-0801029 and DMS-1101373.
Article copyright: © Copyright 2013 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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