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

Lagarias, Jeffrey

doi:10.1090/S0273-0979-2013-01423-X

Euler’s constant: Euler’s work and modern developments
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Bull. Amer. Math. Soc. 50 (2013), 527-628 Request permission

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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