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}
}
  • J. Lagarias
  • Published 7 March 2013
  • Mathematics
  • Bulletin of the American Mathematical Society
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… 
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References

SHOWING 1-10 OF 537 REFERENCES
On a continued fraction expansion for Euler's constant
Euler’s constant, q-logarithms, and formulas of Ramanujan and Gosper
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
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
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
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
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
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
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
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
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…
...
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