Euler's constant

Euler's constant (sometimes called the Euler–Mascheroni constant) is a mathematical constant, usually denoted by the lowercase Greek letter gamma ( $γ$ ), defined as the limiting difference between the harmonic series and the natural logarithm, denoted here by $log$ :

Not to be confused with Euler's number,

e \approx 2.71828

, the base of the natural logarithm.

Euler's constant

Unknown

Analytic number theory

1734

Leonhard Euler

De Progressionibus harmonicis observationes

${\begin{aligned}\gamma &=\lim _{n\to \infty }\left(-\log n+\sum _{k=1}^{n}{\frac {1}{k}}\right)\\[5px]&=\int _{1}^{\infty }\left(-{\frac {1}{x}}+{\frac {1}{\lfloor x\rfloor }}\right)\,\mathrm {d} x.\end{aligned}}$

Here, $⌊\cdot⌋$ represents the floor function.

The numerical value of Euler's constant, to 50 decimal places, is:^[1]

History[edit]

The constant first appeared in a 1734 paper by the Swiss mathematician Leonhard Euler, titled De Progressionibus harmonicis observationes (Eneström Index 43). Euler used the notations $C$ and $O$ for the constant. In 1790, the Italian mathematician Lorenzo Mascheroni used the notations $A$ and $a$ for the constant. The notation $γ$ appears nowhere in the writings of either Euler or Mascheroni, and was chosen at a later time, perhaps because of the constant's connection to the gamma function.^[2] For example, the German mathematician Carl Anton Bretschneider used the notation $γ$ in 1835,^[3] and Augustus De Morgan used it in a textbook published in parts from 1836 to 1842.^[4]

Expressions involving the *

exponential integral

The * of the natural logarithm

Laplace transform

The first term of the expansion for the Riemann zeta function*, where it is the first of the Stieltjes constants*

Laurent series

Calculations of the

digamma function

A product formula for the

gamma function

The asymptotic expansion of the for small arguments.

gamma function

An inequality for

Euler's totient function

The growth rate of the

divisor function

In of Feynman diagrams in quantum field theory

dimensional regularization

The calculation of the

Meissel–Mertens constant

The third of *

Mertens' theorems

Solution of the second kind to

Bessel's equation

In the regularization/ of the harmonic series as a finite value

renormalization

The of the Gumbel distribution

mean

The of the Weibull and Lévy distributions, and, implicitly, of the chi-squared distribution for one or two degrees of freedom.

information entropy

The answer to the *

coupon collector's problem

In some formulations of

Zipf's law

A definition of the *

cosine integral

Lower bounds to a

prime gap

An upper bound on in quantum information theory^[5]

Shannon entropy

for genetics of adaptation in evolutionary biology^[6]

Fisher–Orr model

Bardeen-Cooper-Schrieffer theory of superconductivity (), where it appears as prefactor $2e^{\gamma }/\pi =1.134$ in the BCS equation on the critical temperature.

BCS theory

Euler's constant appears, among other places, in the following (where '*' means that this entry contains an explicit equation):

${\textstyle \gamma \sim H_{n}-\log n-{\frac {1}{2n}}+{\frac {1}{12n^{2}}}-{\frac {1}{120n^{4}}}+\cdots }$ (Euler)

${\textstyle \gamma \sim H_{n}-\log \left({n+{\frac {1}{2}}+{\frac {1}{24n}}-{\frac {1}{48n^{2}}}+\cdots }\right)}$ (Negoi)

${\textstyle \gamma \sim H_{n}-{\frac {\log n+\log(n+1)}{2}}-{\frac {1}{6n(n+1)}}+{\frac {1}{30n^{2}(n+1)^{2}}}-\cdots }$ ()

Cesàro

Bretschneider, Carl Anton (1837) [1835]. . Crelle's Journal (in Latin). 17: 257–285.

"Theoriae logarithmi integralis lineamenta nova"

Havil, Julian (2003). Gamma: Exploring Euler's Constant. Princeton University Press. 978-0-691-09983-5.

ISBN

Lagarias, Jeffrey C. (2013). "Euler's constant: Euler's work and modern developments". . 50 (4): 556. arXiv:1303.1856. doi:10.1090/s0273-0979-2013-01423-x. S2CID 119612431.

Bulletin of the American Mathematical Society

Borwein, Jonathan M.; David M. Bradley; Richard E. Crandall (2000). . Journal of Computational and Applied Mathematics. 121 (1–2): 11. Bibcode:2000JCoAM.121..247B. doi:10.1016/s0377-0427(00)00336-8. Derives $γ$ as sums over Riemann zeta functions.

"Computational Strategies for the Riemann Zeta Function"

Finch, Steven R. (2003). Mathematical Constants. Encyclopedia of Mathematics and its Applications. Vol. 94. Cambridge: Cambridge University Press. 0-521-81805-2.

ISBN

Gerst, I. (1969). "Some series for Euler's constant". Amer. Math. Monthly. 76 (3): 237–275. :10.2307/2316370. JSTOR 2316370.

doi

(1872). "On the history of Euler's constant". Messenger of Mathematics. 1: 25–30. JFM 03.0130.01.

Glaisher, James Whitbread Lee

Gourdon, Xavier; Sebah, P. (2002). .

"Collection of formulae for Euler's constant, $γ$ "

Gourdon, Xavier; Sebah, P. (2004). .

"The Euler constant: $γ$ "

Karatsuba, E. A. (1991). "Fast evaluation of transcendental functions". Probl. Inf. Transm. 27 (44): 339–360.

Karatsuba, E.A. (2000). "On the computation of the Euler constant $γ$ ". Journal of Numerical Algorithms. 24 (1–2): 83–97. :10.1023/A:1019137125281. S2CID 21545868.

doi

(1997). The Art of Computer Programming, Vol. 1 (3rd ed.). Addison-Wesley. pp. 75, 107, 114, 619–620. ISBN 0-201-89683-4.

Knuth, Donald

Lehmer, D. H. (1975). (PDF). Acta Arith. 27 (1): 125–142. doi:10.4064/aa-27-1-125-142.

"Euler constants for arithmetical progressions"

Lerch, M. (1897). "Expressions nouvelles de la constante d'Euler". Sitzungsberichte der Königlich Böhmischen Gesellschaft der Wissenschaften. 42: 5.

(1790). Adnotationes ad calculum integralem Euleri, in quibus nonnulla problemata ab Eulero proposita resolvuntur. Galeati, Ticini.

Mascheroni, Lorenzo

Sondow, Jonathan (2002). "A hypergeometric approach, via linear forms involving logarithms, to irrationality criteria for Euler's constant". Mathematica Slovaca. 59: 307–314. :math.NT/0211075. Bibcode:2002math.....11075S. doi:10.2478/s12175-009-0127-2. S2CID 16340929. with an Appendix by Sergey Zlobin

Euler's constant

Euler's constant

Type

Fields

Discovered

By

First mention

Named after

History[edit]

exponential integral

Laplace transform

Laurent series

digamma function

gamma function

gamma function

Euler's totient function

divisor function

dimensional regularization

Meissel–Mertens constant

Mertens' theorems

Bessel's equation

renormalization

mean

information entropy

coupon collector's problem

Zipf's law

cosine integral

prime gap

Shannon entropy

Fisher–Orr model

BCS theory

Cesàro

"Theoriae logarithmi integralis lineamenta nova"

ISBN

Bulletin of the American Mathematical Society

"Computational Strategies for the Riemann Zeta Function"

ISBN

doi

Glaisher, James Whitbread Lee

"Collection of formulae for Euler's constant, γ"

"The Euler constant: γ"

doi

Knuth, Donald

"Euler constants for arithmetical progressions"

Mascheroni, Lorenzo

arXiv

"Euler constant"

Weisstein, Eric W.

Jonathan Sondow.

Fast Algorithms and the FEE Method

Gourdon and Sebah (2004).

"Collection of formulae for Euler's constant, $γ$ "

"The Euler constant: $γ$ "