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

TL;DR

This paper reviews Euler's foundational work on the constant gamma and explores recent mathematical developments connecting Euler's constant with various advanced topics like the Riemann hypothesis, sieve methods, and transcendence.

Contribution

It provides a comprehensive survey of Euler's work and highlights recent research advances involving Euler's constant and related mathematical constants.

Findings

Connections between Euler's constant and the Riemann hypothesis

Recent results on Diophantine approximation and transcendence

Applications of Euler's constant in sieve methods and random matrix theory

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.