Editing Euler's constant (section)

== Appearances ==
Euler's constant appears frequently in mathematics, especially in [[number theory]] and [[Mathematical analysis|analysis]].<ref>{{Cite web |last=Sondow |first=Jonathan |date=2004 |title=The Euler constant: γ |url=http://numbers.computation.free.fr/Constants/Gamma/gamma.html |access-date=2024-11-01}}</ref> Examples include, among others, the following places: (''where'' ''<nowiki/>'*' means that this entry contains an explicit equation''):

===Analysis===

* The Weierstrass product formula for the [[gamma function]] and the [[Barnes G-function]].<ref name="Davis">{{cite journal |last=Davis |first=P. J. |date=1959 |title=Leonhard Euler's Integral: A Historical Profile of the Gamma Function |url=http://mathdl.maa.org/mathDL/22/?pa=content&sa=viewDocument&nodeId=3104 |url-status=dead |journal=[[American Mathematical Monthly]] |volume=66 |issue=10 |pages=849–869 |doi=10.2307/2309786 |jstor=2309786 |archive-url=https://web.archive.org/web/20121107190256/http://mathdl.maa.org/mathDL/22/?pa=content&sa=viewDocument&nodeId=3104 |archive-date=7 November 2012 |access-date=3 December 2016}}</ref><ref>{{Cite web |title=DLMF: §5.17 Barnes' 𝐺-Function (Double Gamma Function) ‣ Properties ‣ Chapter 5 Gamma Function |url=https://dlmf.nist.gov/5.17 |access-date=2024-11-01 |website=dlmf.nist.gov}}</ref>
* The [[Particular values of the gamma function#General rational argument|asymptotic expansion]] of the gamma function, <math>\Gamma(1/x)\sim x-\gamma</math>.
* Evaluations of the [[digamma function]] at rational values.<ref>{{Cite web |last=Weisstein |first=Eric W. |title=Digamma Function |url=https://mathworld.wolfram.com/DigammaFunction.html |access-date=2024-10-30 |website=mathworld.wolfram.com |language=en}}</ref>
* The [[Laurent series]] expansion for the [[Riemann zeta function]]*, where it is the first of the [[Stieltjes constants]].<ref>{{Cite web |last=Weisstein |first=Eric W. |title=Stieltjes Constants |url=https://mathworld.wolfram.com/StieltjesConstants.html |access-date=2024-11-01 |website=mathworld.wolfram.com |language=en}}</ref>
* Values of the [[Particular values of the Riemann zeta function#Derivatives|derivative of the Riemann zeta function]] and [[Dirichlet beta function#Derivative|Dirichlet beta function]].<ref name=":7" />{{rp|137}}<ref name=":1" />
* In connection to the [[Laplace transform|Laplace]] and [[Mellin transform]].<ref>{{Cite book |last=Williams |first=John |title=Laplace transforms |date=1973 |publisher=Allen & Unwin |isbn=978-0-04-512021-5 |series=Problem solvers |location=London}}</ref><ref>{{Cite web |title=DLMF: §2.5 Mellin Transform Methods ‣ Areas ‣ Chapter 2 Asymptotic Approximations |url=https://dlmf.nist.gov/2.5 |access-date=2024-11-01 |website=dlmf.nist.gov}}</ref>
* In the regularization/[[renormalization]] of the [[harmonic series (mathematics)|harmonic series]] as a finite value.
*Expressions involving the [[exponential integral|exponential]] and [[Logarithmic integral function|logarithmic integral]].*<ref name=":8">{{Cite web |title=DLMF: §6.6 Power Series ‣ Properties ‣ Chapter 6 Exponential, Logarithmic, Sine, and Cosine Integrals |url=https://dlmf.nist.gov/6.6 |access-date=2024-11-01 |website=dlmf.nist.gov}}</ref><ref>{{Cite web |last=Weisstein |first=Eric W. |title=Logarithmic Integral |url=https://mathworld.wolfram.com/LogarithmicIntegral.html |access-date=2024-11-01 |website=mathworld.wolfram.com |language=en}}</ref>
* A definition of the [[trigonometric integral#Cosine integral|cosine integral]].*<ref name=":8" />
* In relation to [[Bessel function|Bessel functions]].<ref>{{Cite web |title=DLMF: §10.32 Integral Representations ‣ Modified Bessel Functions ‣ Chapter 10 Bessel Functions |url=https://dlmf.nist.gov/10.32 |access-date=2024-11-01 |website=dlmf.nist.gov}}</ref><ref>{{Cite web |title=DLMF: §10.22 Integrals ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions |url=https://dlmf.nist.gov/10.22 |access-date=2024-11-01 |website=dlmf.nist.gov}}</ref><ref>{{Cite web |title=DLMF: §10.8 Power Series ‣ Bessel Functions and Hankel Functions ‣ Chapter 10 Bessel Functions |url=https://dlmf.nist.gov/10.8 |access-date=2024-11-01 |website=dlmf.nist.gov}}</ref><ref>{{Cite web |title=DLMF: §10.24 Functions of Imaginary Order ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions |url=https://dlmf.nist.gov/10.24 |access-date=2024-11-01 |website=dlmf.nist.gov}}</ref>
* Asymptotic expansions of modified [[Struve function|Struve functions]].<ref>{{Cite web |title=DLMF: §11.6 Asymptotic Expansions ‣ Struve and Modified Struve Functions ‣ Chapter 11 Struve and Related Functions |url=https://dlmf.nist.gov/11.6 |access-date=2024-11-01 |website=dlmf.nist.gov}}</ref>
* In relation to other [[special functions]].<ref>{{Cite web |title=DLMF: §13.2 Definitions and Basic Properties ‣ Kummer Functions ‣ Chapter 11 Confluent Hypergeometric Functions |url=https://dlmf.nist.gov/13.2 |access-date=2024-11-01 |website=dlmf.nist.gov}}</ref><ref>{{Cite web |title=DLMF: §9.12 Scorer Functions ‣ Related Functions ‣ Chapter 9 Airy and Related Functions |url=https://dlmf.nist.gov/9.12 |access-date=2024-11-01 |website=dlmf.nist.gov}}</ref>

===Number theory===

* An inequality for [[Euler's totient function]].<ref>{{Cite journal |last1=Rosser |first1=J. Barkley |last2=Schoenfeld |first2=Lowell |date=1962 |title=Approximate formulas for some functions of prime numbers |url=https://projecteuclid.org/journals/illinois-journal-of-mathematics/volume-6/issue-1/Approximate-formulas-for-some-functions-of-prime-numbers/10.1215/ijm/1255631807.full |journal=Illinois Journal of Mathematics |volume=6 |issue=1 |pages=64–94 |doi=10.1215/ijm/1255631807 |issn=0019-2082}}</ref>
* The growth rate of the [[divisor function]].<ref>{{Cite book |last1=Hardy |first1=Godfrey H. |title=An introduction to the theory of numbers |last2=Wright |first2=Edward M. |last3=Silverman |first3=Joseph H. |date=2008 |publisher=Oxford University Press |isbn=978-0-19-921986-5 |editor-last=Heath-Brown |editor-first=D. R. |edition=6th |series=Oxford mathematics |location=Oxford New York Auckland |page=469-471}}</ref>
* A formulation of the [[Riemann hypothesis]].<ref name=":2" /><ref>{{Cite journal |last=Robin |first=Guy |date=1984 |title=Grandes valeurs de la fonction somme des diviseurs et hypothèse de Riemann |url=http://zakuski.utsa.edu/~jagy/Robin_1984.pdf |journal=Journal de mathématiques pures et appliquées |volume=63 |pages=187–213}}</ref>
* The third of [[Mertens' theorems]].*<ref name=":9" />
* The calculation of the [[Meissel–Mertens constant]].<ref>{{Cite web |last=Weisstein |first=Eric W. |title=Mertens Constant |url=https://mathworld.wolfram.com/MertensConstant.html |access-date=2024-11-01 |website=mathworld.wolfram.com |language=en}}</ref>
* Lower bounds to specific [[Prime gap#Lower bounds|prime gaps]].<ref>{{Cite journal |last=Pintz |first=János |date=1997-04-01 |title=Very Large Gaps between Consecutive Primes |url=https://www.sciencedirect.com/science/article/pii/S0022314X97920813 |journal=Journal of Number Theory |volume=63 |issue=2 |pages=286–301 |doi=10.1006/jnth.1997.2081 |issn=0022-314X}}</ref>
* An [[approximation]] of the average number of [[Divisor|divisors]] of all numbers from 1 to a given ''n.''<ref name=":6" />
* The [[Lenstra–Pomerance–Wagstaff conjecture]] on the frequency of [[Mersenne prime|Mersenne primes]].<ref>{{Cite web |title=Heuristics: Deriving the Wagstaff Mersenne Conjecture |url=https://t5k.org/mersenne/heuristic.html |access-date=2024-11-01 |website=t5k.org}}</ref>
* An estimation of the efficiency of the [[euclidean algorithm]].<ref>{{Cite web |last=Weisstein |first=Eric W. |title=Porter's Constant |url=https://mathworld.wolfram.com/PortersConstant.html |access-date=2024-11-01 |website=mathworld.wolfram.com |language=en}}</ref>
* Sums involving the [[Möbius function|Möbius]] and [[Von Mangoldt function|von Mangolt function]].
* Estimate of the divisor summatory function of the [[Dirichlet hyperbola method]].<ref>{{Cite book |last=Tenenbaum |first=Gérald |url=https://books.google.de/books?id=UEk-CgAAQBAJ&lpg=PR15&dq=dirichlet%20hyperbola%20method&hl=de&pg=PA360#v=onepage&q=dirichlet%20hyperbola%20method&f=false |title=Introduction to Analytic and Probabilistic Number Theory |date=2015-07-16 |publisher=American Mathematical Soc. |isbn=978-0-8218-9854-3 |language=en}}</ref>

=== In other fields ===

*In some formulations of [[Zipf's law]].
*The answer to the [[coupon collector's problem]].*
* The mean of the [[Gumbel distribution]].
* An approximation of the [[Landau distribution]].
* The [[information entropy]] of the [[Weibull distribution|Weibull]] and [[Lévy distribution|Lévy]] distributions, and, implicitly, of the [[chi-squared distribution]] for one or two degrees of freedom.
* An upper bound on [[Shannon entropy]] in [[quantum information science|quantum information theory]].{{r|CavesFuchs1996}}
* In [[dimensional regularization]] of [[Feynman diagram]]s in [[quantum field theory]].
* In the BCS equation on the critical temperature in [[Bardeen–Cooper–Schrieffer|BCS theory]] of superconductivity.*
* [[Fisher's geometric model|Fisher–Orr model]] for genetics of adaptation in evolutionary biology.{{r|ConnallonHodgins2021}}