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

@article{Lagarias2013EulersCE, title={Euler's constant: Euler's work and modern developments}, author={Jeffrey C. Lagarias}, journal={Bulletin of the American Mathematical Society}, year={2013}, volume={50}, pages={527-628} }

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…

## 131 Citations

Fractional parts and their relations to the values of the Riemann zeta function

- Mathematics
- 2017

A well-known result, due to Dirichlet and later generalized by de la Vallée–Poussin, expresses a relationship between the sum of fractional parts and the Euler–Mascheroni constant. In this paper, we…

On certain alternating series involving zeta and multiple zeta values

- Mathematics
- 2018

In this article, we consider various generalizations of Euler's famous relation
\[
\gamma = \sum_{n\geq 2} (-1)^n \frac{\zeta(n)}{n}
\]
linking Euler's constant $\gamma$ to special values of the…

Euler's factorial series, Hardy integral, and continued fractions

- Mathematics
- 2021

. We study p -adic Euler’s series E p ( t ) = P ∞ k =0 k ! t k at a point p a , a ∈ Z ≥ 1 , and use Pad´e approximations to prove a lower bound for the p -adic absolute value of the expression cE p (…

ON A GENERALIZATION OF EULER’S CONSTANT

- Mathematics
- 2021

A one parameter generalization of Euler’s constant γ from [Numer. Algorithms 46(2) (2007) 141–151] is investigated, and additional expressions for γ are derived. Included are forms involving the…

Shifted Euler constants and a generalization of Euler-Stieltjes constants

- MathematicsJournal of Number Theory
- 2019

Another Alternating Analogue of Euler’s Constant

- MathematicsThe American Mathematical Monthly
- 2021

Abstract In 2005 Jonathan Sondow found several interesting analogies between Euler’s constant γ and revealing that the latter may be regarded as the alternating analogue of Euler’s constant. In this…

From Euler's play with infinite series to the anomalous magnetic moment

- Mathematics
- 2017

During a first St. Petersburg period Leonhard Euler, in his early twenties, became interested in the Basel problem: summing the series of inverse squares (posed by Pietro Mengoli in mid 17th…

Euler ’ s Constant : New Insights by Means of Minus One Factorial

- Mathematics
- 2014

The great object of this paper is to furnish, in a concise and plain manner, new insights into that mysterious constant whose arithmetic nature was shrouded in obscurity for over 250 years, the…

## References

SHOWING 1-10 OF 537 REFERENCES

Euler’s constant, q-logarithms, and formulas of Ramanujan and Gosper

- Mathematics
- 2006

The aim of the paper is to relate computational and arithmetic questions about Euler’s constant γ with properties of the values of the q-logarithm function, with natural choice of~q. By these means,…

Euler's Constant to 1271 Places

- Computer Science
- 1962

The value of Euler's or Mascheroni's constant Pi = lim"'. (1 + 2 + + (1/n) inn) has now been determined to 1271 decimal places, thus extending the previously known value of 328 places. A calculation…

The Theory of the Riemann Zeta-Function

- Mathematics
- 1987

The Riemann zeta-function embodies both additive and multiplicative structures in a single function, making it our most important tool in the study of prime numbers. This volume studies all aspects…

New Vacca-Type Rational Series for Euler's Constant and Its

- Mathematics
- 2005

We recall a pair of logarithmic series that reveals ln(4 ∕ π) to be an “alternating” analog of Euler’s constant γ. Using the binary expansion of an integer, we derive linear, quadratic, and cubic…

Double integrals and infinite products for some classical constants via analytic continuations of Lerch’s transcendent

- Mathematics
- 2008

The two-fold aim of the paper is to unify and generalize on the one hand the double integrals of Beukers for ζ(2) and ζ(3), and of the second author for Euler’s constant γ and its alternating analog…

COMPUTATION OF THE REGULAR CONTINUED FRACTION FOR EULER'S CONSTANT

- Mathematics
- 1977

We describe a computation of the first 20,000 partial quotients in the regular continued fraction fo Euler’s constant = 0.577... and exp( ) = 1.781.... A preliminary step was the calculation of and…

On the arithmetic nature of the values of the gamma function, Euler's constant, and Gompertz's constant

- Mathematics
- 2012

In this paper, we prove some results concerning the arithmetic nature of the values of the Gamma function at rational or algebraic points and for Euler’s constant γ. A (completely open) conjecture of…

On q-analogues of the Euler constant and Lerch's limit formula

- Mathematics, Philosophy
- 2003

We introduce and study a q-analogue γ(q) of the Euler constant via a suitably defined q-analogue of the Riemann zeta function. We show, in particular, that the value γ(2) is irrational. We also…

On the computation of Euler’s constant

- Mathematics
- 1963

The computation of Eider’s constant, γ, to 3566 decimal places by a procedure not previously used is described. As a part of this computation, the natural logarithm of 2 has been evaluated to 3683…