View source for Euler's constant

{{Short description|Difference between logarithm and harmonic series}}
{{distinguish|text = [[Euler's number]], {{math|e ≈ ''2.71828''}}, the base of the natural logarithm}}
{{Use shortened footnotes|date=May 2021}}
{{log(x)}}
{{Infobox mathematical constant
| name = Euler's constant
| symbol = {{mvar|γ}}
| type = Unknown
| fields_of_application = {{flatlist |
* [[Number theory]]
* [[Mathematical analysis|Analysis]]
}}
| approximation = 0.57721...{{r|A001620}}
| discovery_date = 1734
| discovery_person = [[Leonhard Euler]]
| discovery_work = ''De Progressionibus harmonicis observationes''
| named_after = {{flatlist | 
* [[Leonhard Euler]]
* [[Lorenzo Mascheroni]]
}}
}}
[[File:gamma-area.svg|thumb|237px|right|The area of the blue region converges to Euler's constant.]]

'''Euler's constant''' (sometimes called the '''Euler–Mascheroni constant''') is a [[mathematical constant]], usually denoted by the lowercase Greek letter [[gamma]] ({{math|''γ''}}), defined as the [[limit of a sequence|limiting]] difference between the [[harmonic series (mathematics)|harmonic series]] and the [[natural logarithm]], denoted here by {{math|log}}:

<math display="block">\begin{align}
\gamma &= \lim_{n\to\infty}\left(-\log n + \sum_{k=1}^n \frac1{k}\right)\\[5px]
&=\int_1^\infty\left(-\frac1x+\frac1{\lfloor x\rfloor}\right)\,\mathrm dx.
\end{align}</math>

Here, {{math|⌊·⌋}} represents the [[floor and ceiling functions|floor function]].

The numerical value of Euler's constant, to 50 [[Decimal Places|decimal places]], is:{{r|A001620}}

{{block indent|{{gaps|0.57721|56649|01532|86060|65120|90082|40243|10421|59335|93992|...}} }}

== History ==
The constant first appeared in a 1734 paper by the [[Ancien Régime of Switzerland|Swiss]] mathematician [[Leonhard Euler]], titled ''De Progressionibus harmonicis observationes'' (Eneström Index 43), where he described it as "worthy of serious consideration".<ref name=":3" />{{sfn|Lagarias|2013}} Euler initially calculated the constant's value to 6 decimal places. In 1781, he calculated it to 16 decimal places. Euler used the notations {{math|''C''}} and {{math|''O''}} for the constant. The [[Italy|Italian]] mathematician [[Lorenzo Mascheroni]] attempted to calculate the constant to 32 decimal places, but made errors in the 20th–22nd and 31st–32nd decimal places; starting from the 20th digit, he calculated ...'''181'''12090082'''39''' when the correct value is ...'''065'''12090082'''40'''. In 1790, he used the notations {{math|''A''}} and {{math|''a''}} for the constant. Other computations were done by [[Johann Georg von Soldner|Johann von Soldner]] in 1809, who used the notation {{math|''H''}}. The notation {{math|''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]].{{sfn|Lagarias|2013}} For example, the [[Germany|German]] mathematician [[Carl Anton Bretschneider]] used the notation {{math|''γ''}} in 1835,{{sfn|Bretschneider|1837|loc="{{math|1=''γ'' = ''c''}} = {{gaps|0,577215|664901|532860|618112|090082|3...}}" on [https://books.google.com/books?hl=fi&id=OAoPAAAAIAAJ&pg=PA260 p. 260]}} and [[Augustus De Morgan]] used it in a textbook published in parts from 1836 to 1842.{{r|DeMorgan183642}} Euler's constant was also studied by the [[India|Indian]] mathematician [[Srinivasa Ramanujan]] who published one paper on it in 1917.<ref>{{Cite journal |last=Brent |first=Richard P. |date=1994 |title=Ramanujan and Euler's Constant |url=https://maths-people.anu.edu.au/~brent/pd/Euler_CARMA_10.pdf |journal=Proc. Symp. Applied Math. |volume=48 |pages=541–545}}</ref> [[David Hilbert]] mentioned the irrationality of {{math|''γ''}} as an unsolved problem that seems "unapproachable" and, allegedly, the [[England|English]] mathematician [[G. H. Hardy|Godfrey Hardy]] offered to give up his Savilian Chair at [[Oxford]] to anyone who could prove this.<ref name=":3" />

== 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 <math>\Gamma(1/x)</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 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]].

=== 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}}

== Properties ==
===Irrationality and transcendence===
The number {{math|''γ''}} has not been proved [[algebraic number|algebraic]] or [[transcendental number|transcendental]]. In fact, it is not even known whether {{math|''γ''}} is [[irrational number|irrational]]. The ubiquity of {{math|''γ''}} revealed by the large number of equations below and the fact that {{math|{{var|γ}}}} has been called the third most important mathematical constant after [[Pi|{{math|{{var|π}}}}]] and [[E (mathematical constant)|{{math|{{var|e}}}}]]<ref>{{Cite web |title=Eulers Constant |url=https://num.math.uni-goettingen.de/~skraemer/gamma.html |access-date=2024-10-19 |website=num.math.uni-goettingen.de}}</ref><ref name=":7">{{Cite book |last=Finch |first=Steven R. |url=https://books.google.com/books?id=Pl5I2ZSI6uAC |title=Mathematical Constants |date=2003-08-18 |publisher=Cambridge University Press |isbn=978-0-521-81805-6 |language=en}}</ref> makes the irrationality of {{math|''γ''}} a major open question in mathematics.<ref name=":3" /><ref>{{Cite web |last=Waldschmidt |first=Michel |date=2023 |title=Some of the most famous open problems in number theory |url=https://webusers.imj-prg.fr/~michel.waldschmidt/articles/pdf/OpenPbsNT.pdf}}</ref>{{r|Sondow2003a}}<ref name=":6">{{Cite book |last1=Conway |first1=John H. |url=https://books.google.com/books?id=0--3rcO7dMYC |title=The Book of Numbers |last2=Guy |first2=Richard |date=1998-03-16 |publisher=Springer Science & Business Media |isbn=978-0-387-97993-9 |language=en}}</ref>

{{unsolved|mathematics|Is Euler's constant irrational? If so, is it transcendental?}}

However, some progress has been made. In 1959 Andrei Shidlovsky proved that at least one of Euler's constant {{math|''γ''}} and the [[Euler–Gompertz constant|Gompertz constant]] {{math|''δ''}} is irrational;{{r|Aptekarev2009}}<ref name=":2">{{Cite web |last=Waldschmidt |first=Michel |date=2023 |title=On Euler's Constant |url=https://webusers.imj-prg.fr/~michel.waldschmidt/articles/pdf/EulerConstant.pdf |place=Sorbonne Université, Institut de Mathématiques de Jussieu, Paris}}</ref> [[Tanguy Rivoal]] proved in 2012 that at least one of them is transcendental.{{r|Rivoal2012}} [[Kurt Mahler]] showed in 1968 that the number <math>\tfrac{\pi}{2}\tfrac{Y_0(2)}{J_0(2)}-\gamma</math> is transcendental (with <math>J_\alpha(x)</math> and <math>Y_\alpha(x)</math> being [[Bessel function]]s).{{r|Mahler1968}}{{sfn|Lagarias|2013}} It is known that the [[Transcendental extension|transcendence degree]] of the field <math>\mathbb Q(e,\gamma,\delta)</math> is at least two.{{sfn|Lagarias|2013}} In 2010 [[M. Ram Murty]] and N. Saradha showed that at most one of the Euler-Lehmer constants, a. i. the numbers of the form

<math display="block">\gamma(a,q) = \lim_{n\rightarrow\infty}\left( - \frac{\log{(a+nq})}{q} + \sum_{k=0}^n{\frac{1}{a+kq}}\right)</math>

is algebraic, given that {{math|{{var|q}} ≥ 2}} and {{math|1 ≤ {{var|a}} < {{var|q}}}}; this family includes the special case {{math|1={{var|γ}}(2,4) = {{sfrac|{{var|γ}}|4}}}}.{{sfn|Lagarias|2013}}{{r|RamMurtySaradha2010}} In 2013 M. Ram Murty and A. Zaytseva found a different family containing {{mvar|γ}}, which is based on sums of reciprocals of integers not divisible by a fixed list of primes, with the same property.{{sfn|Lagarias|2013}}{{r|MurtyZaytseva2013}}

Using a [[continued fraction]] analysis, Papanikolaou showed in 1997 that if {{math|''γ''}} is [[rational number|rational]], its denominator must be greater than 10<sup>244663</sup>.{{r|HaiblePapanikolaou1998|Papanikolaou1997}} If {{math|{{var|e}}{{i sup|1px|{{var|γ}}}}}} is a rational number, then its denominator must be greater than 10<sup>15000</sup>.{{sfn|Lagarias|2013}}

Euler's constant is conjectured not to be an [[Period (algebraic geometry)|algebraic period]],{{sfn|Lagarias|2013}} but the values of its first 10<sup>9</sup> decimal digits seem to indicate that it could be a [[normal number]].<ref>{{Cite web |last=Weisstein |first=Eric W. |title=Euler-Mascheroni Constant Digits |url=https://mathworld.wolfram.com/Euler-MascheroniConstantDigits.html |access-date=2024-10-19 |website=mathworld.wolfram.com |language=en}}</ref>

=== Continued fraction ===
The simple [[continued fraction]] expansion of Euler's constant is given by:{{r|OEIS_A002852}}

:<math>\gamma=0+\cfrac{1}{1+\cfrac{1}{1+\cfrac{1}{2+\cfrac{1}{1+\cfrac{1}{2+\cfrac{1}{1+\cfrac{1}{4+\dots}}}}}}}</math>

which has no ''apparent'' pattern. It is known to have at least 16,695,000,000 terms,{{r|OEIS_A002852}} and it has infinitely many terms [[if and only if]] {{mvar|γ}} is irrational.

[[File:KhinchinBeispiele.svg|thumb|The Khinchin limits for <math>\pi</math> (red), <math>\gamma</math> (blue) and <math>\sqrt[3]{2}</math> (green).|350x350px]]

Numerical evidence suggests that both Euler's constant {{math|{{var|γ}}}} as well as the constant {{math|{{var|e}}{{i sup|1px|{{var|γ}}}}}} are among the numbers for which the [[geometric mean]] of their simple continued fraction terms converges to [[Khinchin's constant]]. Similarly, when <math>p_n/q_n</math> are the convergents of their respective continued fractions, the limit <math>\lim_{n\to\infty}q_n^{1/n}</math> appears to converge to [[Lévy's constant]] in both cases.<ref name=":4">{{Cite journal |last=Brent |first=Richard P. |date=1977 |title=Computation of the Regular Continued Fraction for Euler's Constant |url=https://www.jstor.org/stable/2006010 |journal=Mathematics of Computation |volume=31 |issue=139 |pages=771–777 |doi=10.2307/2006010 |jstor=2006010 |issn=0025-5718}}</ref> However neither of these limits has been proven.<ref name=":0">{{Cite web |last=Weisstein |first=Eric W. |title=Euler-Mascheroni Constant Continued Fraction |url=https://mathworld.wolfram.com/Euler-MascheroniConstantContinuedFraction.html |access-date=2024-09-23 |website=mathworld.wolfram.com |language=en}}</ref>

There also exists a generalized continued fraction for Euler's constant.<ref>{{Citation |last1=Pilehrood |first1=Khodabakhsh Hessami |title=On a continued fraction expansion for Euler's constant |date=2013-12-29 |last2=Pilehrood |first2=Tatiana Hessami|arxiv=1010.1420 }}</ref>

A good simple [[approximation]] of {{math|{{var|γ}}}} is given by the [[Multiplicative inverse|reciprocal]] of the [[square root of 3]] or about 0.57735:<ref>{{Cite web |last=Weisstein |first=Eric W. |title=Euler-Mascheroni Constant Approximations |url=https://mathworld.wolfram.com/Euler-MascheroniConstantApproximations.html |access-date=2024-10-19 |website=mathworld.wolfram.com |language=en}}</ref>

:<math>\frac1\sqrt {3}=0+\cfrac{1}{1+\cfrac{1}{1+\cfrac{1}{2+\cfrac{1}{1+\cfrac{1}{2+\cfrac{1}{1+\cfrac{1}{2+\dots}}}}}}}</math>

with the difference being about 1 in 7,429.

== Formulas and identities ==

=== Relation to gamma function ===
{{mvar|γ}} is related to the [[digamma function]] {{math|Ψ}}, and hence the [[derivative]] of the [[gamma function]] {{math|Γ}}, when both functions are evaluated at 1. Thus:

<math display="block">-\gamma = \Gamma'(1) = \Psi(1). </math>

This is equal to the limits:

<math display="block">\begin{align}-\gamma &= \lim_{z\to 0}\left(\Gamma(z) - \frac1{z}\right) \\&= \lim_{z\to 0}\left(\Psi(z) + \frac1{z}\right).\end{align}</math>

Further limit results are:{{r|Krämer2005}}

<math display="block">\begin{align} \lim_{z\to 0}\frac1{z}\left(\frac1{\Gamma(1+z)} - \frac1{\Gamma(1-z)}\right) &= 2\gamma \\
\lim_{z\to 0}\frac1{z}\left(\frac1{\Psi(1-z)} - \frac1{\Psi(1+z)}\right) &= \frac{\pi^2}{3\gamma^2}. \end{align}</math>

A limit related to the [[beta function]] (expressed in terms of [[gamma function]]s) is

<math display="block">\begin{align} \gamma &= \lim_{n\to\infty}\left(\frac{ \Gamma\left(\frac1{n}\right) \Gamma(n+1)\, n^{1+\frac1{n}}}{\Gamma\left(2+n+\frac1{n}\right)} - \frac{n^2}{n+1}\right) \\
&= \lim\limits_{m\to\infty}\sum_{k=1}^m{m \choose k}\frac{(-1)^k}{k}\log\big(\Gamma(k+1)\big). \end{align}</math>

=== Relation to the zeta function ===
{{mvar|γ}} can also be expressed as an [[series (mathematics)|infinite sum]] whose terms involve the [[Riemann zeta function]] evaluated at positive integers:

<math display="block">\begin{align}\gamma &= \sum_{m=2}^{\infty} (-1)^m\frac{\zeta(m)}{m} \\
 &= \log\frac4{\pi} + \sum_{m=2}^{\infty} (-1)^m\frac{\zeta(m)}{2^{m-1}m}.\end{align} </math>
The constant <math>\gamma</math> can also be expressed in terms of the sum of the reciprocals of [[Riemann hypothesis|non-trivial zeros]] <math>\rho</math> of the zeta function:<ref name="Marek6infinity2019">{{Cite arXiv
| last = Wolf
| first = Marek
| title = 6+infinity new expressions for the Euler-Mascheroni constant
| year = 2019
| eprint = 1904.09855
| class = math.NT
| quote = "The above sum is real and convergent when zeros <math>\rho</math> and complex conjugate <math>\bar{\rho}</math> are paired together and summed according to increasing absolute values of the imaginary parts of {{nowrap|<math>\rho</math>.}}"}} See formula 11 on page 3. Note the typographical error in the numerator of Wolf's sum over zeros, which should be 2 rather than 1.</ref>
:<math>\gamma = \log 4\pi + \sum_{\rho} \frac{2}{\rho} - 2</math>
Other series related to the zeta function include:

<math display="block">\begin{align} \gamma &= \tfrac3{2}- \log 2 - \sum_{m=2}^\infty (-1)^m\,\frac{m-1}{m}\big(\zeta(m)-1\big) \\
 &= \lim_{n\to\infty}\left(\frac{2n-1}{2n} - \log n + \sum_{k=2}^n \left(\frac1{k} - \frac{\zeta(1-k)}{n^k}\right)\right) \\
 &= \lim_{n\to\infty}\left(\frac{2^n}{e^{2^n}} \sum_{m=0}^\infty \frac{2^{mn}}{(m+1)!} \sum_{t=0}^m \frac1{t+1} - n \log 2+ O \left (\frac1{2^{n}\, e^{2^n}}\right)\right).\end{align}</math>

The error term in the last equation is a rapidly decreasing function of {{mvar|n}}. As a result, the formula is well-suited for efficient computation of the constant to high precision.

Other interesting limits equaling Euler's constant are the antisymmetric limit:{{r|Sondow1998}}

<math display="block">\begin{align} \gamma &= \lim_{s\to 1^+}\sum_{n=1}^\infty \left(\frac1{n^s}-\frac1{s^n}\right) \\&= \lim_{s\to 1}\left(\zeta(s) - \frac{1}{s-1}\right) \\&= \lim_{s\to 0}\frac{\zeta(1+s)+\zeta(1-s)}{2} \end{align}</math>

and the following formula, established in 1898 by [[Charles Jean de la Vallée-Poussin|de la Vallée-Poussin]]:

<math display="block">\gamma = \lim_{n\to\infty}\frac1{n}\, \sum_{k=1}^n \left(\left\lceil \frac{n}{k} \right\rceil - \frac{n}{k}\right)</math>

where {{math|{{ceil|&nbsp;}}}} are [[ceiling function|ceiling]] brackets.
This formula indicates that when taking any positive integer {{mvar|n}} and dividing it by each positive integer {{mvar|k}} less than {{mvar|n}}, the average fraction by which the quotient {{math|{{var|n}}/{{var|k}}}} falls short of the next integer tends to {{mvar|γ}} (rather than 0.5) as {{mvar|n}} tends to infinity.

Closely related to this is the [[rational zeta series]] expression. By taking separately the first few terms of the series above, one obtains an estimate for the classical series limit:

<math display="block">\gamma =\lim_{n\to\infty}\left( \sum_{k=1}^n \frac1{k} - \log n -\sum_{m=2}^\infty \frac{\zeta(m,n+1)}{m}\right),</math>

where {{math|{{var|ζ}}({{var|s}}, {{var|k}})}} is the [[Hurwitz zeta function]]. The sum in this equation involves the [[harmonic number]]s, {{math|{{var|H}}{{sub|{{var|n}}}}}}. Expanding some of the terms in the Hurwitz zeta function gives:

<math display="block">H_n = \log(n) + \gamma + \frac1{2n} - \frac1{12n^2} + \frac1{120n^4} - \varepsilon,</math>
where {{math|0 < {{var|ε}} < {{sfrac|1|252{{var|n}}{{sup|6}}}}.}}

{{mvar|γ}} can also be expressed as follows where {{mvar|A}} is the [[Glaisher–Kinkelin constant]]:

<math display="block">\gamma =12\,\log(A)-\log(2\pi)+\frac{6}{\pi^2}\,\zeta'(2)</math>

{{mvar|γ}} can also be expressed as follows, which can be proven by expressing the [[Riemann zeta function|zeta function]] as a [[Laurent series]]:

<math display="block">\gamma=\lim_{n\to\infty}\left(-n+\zeta\left(\frac{n+1}{n}\right)\right)</math>
=== Relation to triangular numbers ===
Numerous formulations have been derived that express <math>\gamma</math> in terms of sums and logarithms of [[triangular numbers]].<ref name="Boya2008AnotherRelation">{{Cite journal
| last = Boya
| first = L.J.
| title = Another relation between π, e, γ and ζ(n)
| journal = Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas
| volume = 102
| pages = 199–202
| year = 2008
| issue = 2
| url = https://doi.org/10.1007/BF03191819
| doi = 10.1007/BF03191819
| quote = "γ/2 in (10) reflects the residual (finite part) of ζ(1)/2, of course."
}} See formulas 1 and 10.</ref><ref name="Jonathan2005DoubleIntegrals">{{Cite journal
| last = Sondow
| first = Jonathan
| title = Double Integrals for Euler's Constant and <math>\textstyle \frac{4}{\pi}</math> and an Analog of Hadjicostas's Formula
| journal = The American Mathematical Monthly
| volume = 112
| issue = 1
| year = 2005
| pages = 61–65
| url = https://doi.org/10.2307/30037385
| doi = 10.2307/30037385
| jstor = 30037385
| access-date = 2024-04-27
}}</ref><ref>{{Cite journal
| last = Chen
| first = Chao-Ping
| title = Ramanujan's formula for the harmonic number
| journal = Applied Mathematics and Computation
| volume = 317
| year = 2018
| pages = 121–128
| issn = 0096-3003
| doi = 10.1016/j.amc.2017.08.053
| url = https://www.sciencedirect.com/science/article/pii/S0096300317306112
| access-date = 2024-04-27
}}</ref><ref>{{cite journal
 | last = Lodge
 | first = A.
 | title = An approximate expression for the value of 1 + 1/2 + 1/3 + ... + 1/r
 | journal = Messenger of Mathematics
 | volume = 30
 | year = 1904
 | pages = 103–107
 | url = https://books.google.com/books?id=K4daAAAAYAAJ&dq=%22An%20approximate%20expression%20for%20the%20value%20of%201%2B%22&pg=PA103
}}</ref> One of the earliest of these is a formula<ref>{{Cite arXiv
| last = Villarino
| first = Mark B.
| title = Ramanujan's Harmonic Number Expansion into Negative Powers of a Triangular Number
| year = 2007
| eprint = 0707.3950
| class = math.CA
| quote = It would also be interesting to develop an expansion for n! into powers of m, a new ''Stirling'' expansion, as it were.
}} See formula 1.8 on page 3.</ref><ref>{{Cite journal
| last = Mortici
| first = Cristinel
| year = 2010
| title = On the Stirling expansion into negative powers of a triangular number
| journal = Math. Commun.
| volume = 15
| pages = 359–364
| url = https://www.researchgate.net/publication/228562533
| doi =
}}</ref> for the {{nowrap|<math>n</math>th}} [[harmonic number]] attributed to [[Srinivasa Ramanujan]] where <math>\gamma</math> is related to <math>\textstyle \ln 2T_{k}</math> in a series that considers the powers of <math>\textstyle \frac{1}{T_{k}}</math> (an earlier, less-generalizable proof<ref>{{Cite journal |last=Cesàro |first=E. |title=Sur la série harmonique |journal=Nouvelles annales de mathématiques: Journal des candidats aux écoles polytechnique et normale |volume=4 |pages=295–296 |year=1885 |url=http://eudml.org/doc/100057 |language=fr |publisher=Carilian-Goeury et Vor Dalmont}}</ref><ref>{{cite book
 | last = Bromwich
 | first = Thomas John I'Anson
 | title = An Introduction to the Theory of Infinite Series
 | publisher = American Mathematical Society
 | year = 2005
 | origyear = 1908
 | edition = 3rd
 | location = United Kingdom
 | url = https://www.dbraulibrary.org.in/RareBooks/An%20introduction%20to%20the%20theory%20of%20infinite%20series.pdf
 | page = 460
}} See exercise 18.</ref> by [[Ernesto Cesàro]] gives the first two terms of the series, with an error term):

:<math>\begin{align} 
    \gamma  
    &= H_u - \frac{1}{2} \ln 2T_u - \sum_{k=1}^{v}\frac{R(k)}{T_{u}^{k}}-\Theta_{v}\,\frac{R(v+1)}{T_{u}^{v+1}}
\end{align}</math>

From [[Stirling's approximation]]<ref name="Boya2008AnotherRelation"/><ref>{{cite book
 | last1 = Whittaker
 | first1 = E.
 | last2 = Watson
 | first2 = G.
 | title = A Course of Modern Analysis
 | edition = 5th
 | origyear = 1902
 | year = 2021
 | page = 271, 275
 | isbn = 9781316518939
 | doi = 10.1017/9781009004091
}} See Examples 12.21 and 12.50 for exercises on the derivation of the integral form <math>\textstyle \int_{-1}^{0} \ln\Gamma(z+1)\,dz</math> of the series <math>\textstyle \sum_{k=1}^{n} \frac{\zeta(k)}{110_{k}} = \ln(\sqrt{2\pi})</math>.</ref> follows a similar series:

:<math>\gamma = \ln 2\pi - \sum_{k=2}^{\infty} \frac{\zeta(k)}{T_{k}}</math>

The series of inverse triangular numbers also features in the study of the [[Basel problem]]{{sfn|Lagarias|2013|p=13}}<ref>{{cite journal |last=Nelsen |first=R. B. |title=Proof without Words: Sum of Reciprocals of Triangular Numbers |journal=Mathematics Magazine |volume=64 |issue=3 |year=1991 |pages=167|doi=10.1080/0025570X.1991.11977600 }}</ref> posed by [[Pietro Mengoli]]. Mengoli proved that <math>\textstyle \sum_{k = 1}^\infty \frac{1}{2T_k} = 1</math>, a result [[Jacob Bernoulli]] later used to estimate the [[Basel_problem#The_Riemann_zeta_function|value]] of <math>\zeta(2)</math>, placing it between <math>1</math> and <math>\textstyle \sum_{k = 1}^\infty \frac{2}{2T_k} = \sum_{k = 1}^\infty \frac{1}{T_{k}} = 2</math>. This identity appears in a formula used by [[Bernhard Riemann]] to compute [[Eulers_constant#Relation_to_the_zeta_function|roots of the zeta function]],<ref>{{Cite book | last = Edwards | first = H. M. | title = Riemann's Zeta Function | publisher = Academic Press | year = 1974 | series = Pure and Applied Mathematics, Vol. 58 | pages = 67, 159}}</ref> where <math>\gamma</math> is expressed in terms of the sum of roots <math>\rho</math> plus the difference between Boya's expansion and the series of exact [[Unit fraction|unit fractions]] <math>\textstyle \sum_{k = 1}^{\infty} \frac{1}{T_{k}}</math>:

:<math>\gamma - \ln 2 = \ln 2\pi + \sum_{\rho} \frac{2}{\rho} - \sum_{k = 1}^{\infty} \frac{1}{T_k}</math>

=== Integrals ===
{{mvar|γ}} equals the value of a number of definite [[integral]]s:

<math display="block">\begin{align}
\gamma &= - \int_0^\infty e^{-x} \log x \,dx \\
 &= -\int_0^1\log\left(\log\frac 1 x \right) dx \\ 
 &= \int_0^\infty \left(\frac1{e^x-1}-\frac1{x\cdot e^x} \right)dx \\
 &= \int_0^1\frac{1-e^{-x}}{x} \, dx -\int_1^\infty \frac{e^{-x}}{x}\, dx\\
 &= \int_0^1\left(\frac1{\log x} + \frac1{1-x}\right)dx\\
 &= \int_0^\infty \left(\frac1{1+x^k}-e^{-x}\right)\frac{dx}{x},\quad k>0\\
 &= 2\int_0^\infty \frac{e^{-x^2}-e^{-x}}{x} \, dx ,\\
&= \log\frac{\pi}{4}-\int_0^\infty \frac{\log x}{\cosh^2x} \, dx ,\\
 &= \int_0^1 H_x \, dx, \\
 &= \frac{1}{2}+\int_{0}^{\infty}\log\left(1+\frac{\log\left(1+\frac{1}{t}\right)^{2}}{4\pi^{2}}\right)dt \\
&= 1-\int_0^1 \{1/x\} dx 
 \end{align}
 </math>
where {{math|{{var|H}}{{sub|{{var|x}}}}}} is the [[Harmonic number#Harmonic numbers for real and complex values|fractional harmonic number]], and <math>\{1/x\}</math> is the [[fractional part]] of <math>1/x</math>.

The third formula in the integral list can be proved in the following way:

<math display="block">\begin{align}
&\int_0^{\infty} \left(\frac{1}{e^x - 1} - \frac{1}{x e^x} \right) dx
 = \int_0^{\infty} \frac{e^{-x} + x - 1}{x[e^x -1]} dx
 = \int_0^{\infty} \frac{1}{x[e^x - 1]} \sum_{m = 1}^{\infty} \frac{(-1)^{m+1}x^{m+1}}{(m+1)!} dx \\[2pt]
&= \int_0^{\infty} \sum_{m = 1}^{\infty} \frac{(-1)^{m+1}x^m}{(m+1)![e^x -1]} dx
 = \sum_{m = 1}^{\infty} \int_0^{\infty} \frac{(-1)^{m+1}x^m}{(m+1)![e^x -1]} dx
 = \sum_{m = 1}^{\infty} \frac{(-1)^{m+1}}{(m+1)!} \int_0^{\infty} \frac{x^m}{e^x - 1} dx \\[2pt]
&= \sum_{m = 1}^{\infty} \frac{(-1)^{m+1}}{(m+1)!} m!\zeta(m+1)
 = \sum_{m = 1}^{\infty} \frac{(-1)^{m+1}}{m+1}\zeta(m+1)
 = \sum_{m = 1}^{\infty} \frac{(-1)^{m+1}}{m+1} \sum_{n = 1}^{\infty}\frac{1}{n^{m+1}}
 = \sum_{m = 1}^{\infty} \sum_{n = 1}^{\infty} \frac{(-1)^{m+1}}{m+1}\frac{1}{n^{m+1}} \\[2pt]
&= \sum_{n = 1}^{\infty} \sum_{m = 1}^{\infty} \frac{(-1)^{m+1}}{m+1}\frac{1}{n^{m+1}}
 = \sum_{n = 1}^{\infty} \left[\frac{1}{n} - \log\left(1+\frac{1}{n}\right)\right]
 = \gamma
\end{align}</math>

The integral on the second line of the equation stands for the [[Debye function]] value of {{math|+∞}}, which is {{math|{{var|m}}!{{var|ζ}}({{var|m}} + 1)}}.

Definite integrals in which {{mvar|γ}} appears include:<ref name=":3">{{Cite web |last=Weisstein |first=Eric W. |title=Euler-Mascheroni Constant |url=https://mathworld.wolfram.com/Euler-MascheroniConstant.html |access-date=2024-10-19 |website=mathworld.wolfram.com |language=en}}</ref><ref name=":1">{{Cite journal |last=Blagouchine |first=Iaroslav V. |date=2014-10-01 |title=Rediscovery of Malmsten's integrals, their evaluation by contour integration methods and some related results |url=https://link.springer.com/article/10.1007/s11139-013-9528-5 |journal=The Ramanujan Journal |language=en |volume=35 |issue=1 |pages=21–110 |doi=10.1007/s11139-013-9528-5 |issn=1572-9303}}</ref>

<math display="block">\begin{align}
\int_0^\infty e^{-x^2} \log x \,dx &= -\frac{(\gamma+2\log 2)\sqrt{\pi}}{4} \\
\int_0^\infty e^{-x} \log^2 x \,dx &= \gamma^2 + \frac{\pi^2}{6}
\\ \int_0^\infty \frac{e^{-x}\log x}{e^x +1} \,dx &= \frac12 \log^2 2 - \gamma \end{align}</math>

We also have [[Eugène Charles Catalan|Catalan]]'s 1875 integral{{r|SondowZudilin2006}}

<math display="block">\gamma = \int_0^1 \left(\frac1{1+x}\sum_{n=1}^\infty x^{2^n-1}\right)\,dx.</math>

One can express {{mvar|γ}} using a special case of [[Hadjicostas's formula]] as a [[Multiple integral#Double integral|double integral]]{{r|Sondow2003a|Sondow2005}} with equivalent series:

<math display="block">\begin{align}
\gamma &= \int_0^1 \int_0^1 \frac{x-1}{(1-xy)\log xy}\,dx\,dy \\
&= \sum_{n=1}^\infty \left(\frac 1 n -\log\frac{n+1} n \right).
\end{align}</math>

An interesting comparison by Sondow{{r|Sondow2005}} is the double integral and alternating series

<math display="block">\begin{align}
\log\frac 4 \pi &= \int_0^1 \int_0^1 \frac{x-1}{(1+xy)\log xy} \,dx\,dy \\
&= \sum_{n=1}^\infty \left((-1)^{n-1}\left(\frac 1 n -\log\frac{n+1} n \right)\right).
\end{align}</math>

It shows that {{math|log {{sfrac|4|π}}}} may be thought of as an "alternating Euler constant".

The two constants are also related by the pair of series{{r|Sondow2005a}}

<math display="block">\begin{align}
\gamma &= \sum_{n=1}^\infty \frac{N_1(n) + N_0(n)}{2n(2n+1)} \\
\log\frac4{\pi} &= \sum_{n=1}^\infty \frac{N_1(n) - N_0(n)}{2n(2n+1)} ,
\end{align}</math>

where {{math|{{var|N}}{{sub|1}}({{var|n}})}} and {{math|{{var|N}}{{sub|0}}({{var|n}})}} are the number of 1s and 0s, respectively, in the [[Binary number|base 2]] expansion of {{mvar|n}}.

=== Series expansions ===
In general,

<math display="block">
\gamma = \lim_{n \to \infty}\left(\frac{1}{1}+\frac{1}{2}+\frac{1}{3} + \ldots + \frac{1}{n} - \log(n+\alpha) \right) \equiv \lim_{n \to \infty} \gamma_n(\alpha)
</math>

for any {{math|{{var|α}} > −{{var|n}}}}. However, the rate of convergence of this expansion depends significantly on {{mvar|α}}. In particular, {{math|{{var|γ}}{{sub|{{var|n}}}}(1/2)}} exhibits much more rapid convergence than the conventional expansion {{math|{{var|γ}}{{sub|{{var|n}}}}(0)}}.{{r|DeTemple1993}}{{sfn|Havil|2003|pp=75–8}} This is because

<math display="block">
\frac{1}{2(n+1)} < \gamma_n(0) - \gamma < \frac{1}{2n},
</math>

while

<math display="block">
\frac{1}{24(n+1)^2} < \gamma_n(1/2) - \gamma < \frac{1}{24n^2}.
</math>

Even so, there exist other series expansions which converge more rapidly than this; some of these are discussed below.

Euler showed that the following [[infinite series]] approaches {{mvar|γ}}:
<math display="block">\gamma = \sum_{k=1}^\infty \left(\frac 1 k - \log\left(1+\frac 1 k \right)\right).</math>

The series for {{mvar|γ}} is equivalent to a series [[Niels Nielsen (mathematician)|Nielsen]] found in 1897:{{r|Krämer2005}}{{sfn|Blagouchine|2016}}

<math display="block">\gamma = 1 - \sum_{k=2}^\infty (-1)^k\frac{\left\lfloor\log_2 k\right\rfloor}{k+1}.</math>

In 1910, [[Giovanni Vacca (mathematician)|Vacca]] found the closely related series{{r|Vacca1910|Glaisher1910|Hardy1912|Vacca1926|Kluyver1927|Krämer2005|Blagouchine2016}}

<math display="block">\begin{align}
\gamma & = \sum_{k=1}^\infty (-1)^k\frac{\left\lfloor\log_2 k\right\rfloor} k \\[5pt]
& = \tfrac12-\tfrac13 + 2\left(\tfrac14 - \tfrac15 + \tfrac16 - \tfrac17\right) + 3\left(\tfrac18 - \tfrac19 + \tfrac1{10} - \tfrac1{11} + \cdots - \tfrac1{15}\right) + \cdots,
\end{align}</math>

where {{math|log{{sub|2}}}} is the [[binary logarithm|logarithm to base 2]] and {{math|{{floor|&nbsp;}}}} is the [[Floor and ceiling functions|floor function]].

This can be generalized to:<ref>{{Citation |last1=Pilehrood |first1=Khodabakhsh Hessami |title=Vacca-type series for values of the generalized-Euler-constant function and its derivative |date=2008-08-04 |last2=Pilehrood |first2=Tatiana Hessami|arxiv=0808.0410 }}</ref>

<math display="block">\gamma= \sum_{k=1}^\infty \frac{\left\lfloor\log_B k\right\rfloor}{k} \varepsilon(k)</math>where:<math display="block">\varepsilon(k)= \begin{cases} B-1, &\text{if } B\mid n \\ -1, &\text{if }B\nmid n \end{cases}</math>

In 1926 Vacca found a second series:

<math display="block">\begin{align}
\gamma + \zeta(2) & = \sum_{k=2}^\infty \left( \frac1{\left\lfloor\sqrt{k}\right\rfloor^2} - \frac1{k}\right) \\[5pt]
& = \sum_{k=2}^\infty \frac{k - \left\lfloor\sqrt{k}\right\rfloor^2}{k \left\lfloor \sqrt{k} \right\rfloor^2} \\[5pt]
&= \frac12 + \frac23 + \frac1{2^2}\sum_{k=1}^{2\cdot 2} \frac{k}{k+2^2} + \frac1{3^2}\sum_{k=1}^{3\cdot 2} \frac{k}{k+3^2} + \cdots
\end{align}</math>

From the [[Carl Johan Malmsten|Malmsten]]–[[Ernst Kummer|Kummer]] expansion for the logarithm of the gamma function<ref name=":1" /> we get:

<math display="block">\gamma = \log\pi - 4\log\left(\Gamma(\tfrac34)\right) + \frac 4 \pi \sum_{k=1}^\infty (-1)^{k+1}\frac{\log(2k+1)}{2k+1}.</math>

Ramanujan, in his [[Ramanujan's lost notebook|lost notebook]] gave a series that approaches {{mvar|γ}}{{r|Berndt2008}}:

<math display="block">\gamma = \log 2 - \sum_{n=1}^{\infty} \sum_{k=\frac{3^{n-1}+1}{2}}^{\frac{3^{n}-1}{2}} \frac{2n}{(3k)^3-3k}</math>

An important expansion for Euler's constant is due to [[Gregorio Fontana|Fontana]] and [[Lorenzo Mascheroni|Mascheroni]]

<math display="block">\gamma = \sum_{n=1}^\infty \frac{|G_n|}{n} = \frac12 + \frac1{24} + \frac1{72} + \frac{19}{2880} + \frac3{800} + \cdots,</math>
where {{math|{{var|G}}{{sub|{{var|n}}}}}} are [[Gregory coefficients]].{{r|Krämer2005|Blagouchine2016|Blagouchine2018}} This series is the special case {{math|1={{var|k}} = 1}} of the expansions

<math display="block">\begin{align}
 \gamma &= H_{k-1}  - \log k + \sum_{n=1}^{\infty}\frac{(n-1)!|G_n|}{k(k+1) \cdots (k+n-1)} && \\
     &= H_{k-1} - \log k + \frac1{2k} + \frac1{12k(k+1)} + \frac1{12k(k+1)(k+2)} + \frac{19}{120k(k+1)(k+2)(k+3)} + \cdots &&
\end{align}</math>

convergent for {{math|1={{var|k}} = 1, 2, ...}}

A similar series with the Cauchy numbers of the second kind {{math|{{var|C}}{{sub|{{var|n}}}}}} is{{r|Blagouchine2016|Alabdulmohsin2018_1478}} 

<math display="block">\gamma = 1 -  \sum_{n=1}^\infty \frac{C_{n}}{n \, (n+1)!} =1- \frac{1}{4} -\frac{5}{72} - \frac{1}{32} - \frac{251}{14400} - \frac{19}{1728} - \ldots</math>

Blagouchine (2018) found an interesting generalisation of the Fontana–Mascheroni series

<math display="block">\gamma=\sum_{n=1}^\infty\frac{(-1)^{n+1}}{2n}\Big\{\psi_{n}(a)+ \psi_{n}\Big(-\frac{a}{1+a}\Big)\Big\},
\quad a>-1</math>

where {{math|{{var|ψ}}{{sub|{{var|n}}}}({{var|a}})}} are the [[Bernoulli polynomials of the second kind]], which are defined by the generating function

<math display="block">
\frac{z(1+z)^s}{\log(1+z)}= \sum_{n=0}^\infty z^n \psi_n(s) ,\qquad |z|<1.
</math>

For any rational {{mvar|a}} this series contains rational terms only. For example, at {{math|1={{var|a}} = 1}}, it becomes{{r|OEIS_A302120|OEISA302121}}

<math display="block">\gamma=\frac{3}{4} - \frac{11}{96} - \frac{1}{72} - \frac{311}{46080} - \frac{5}{1152} - \frac{7291}{2322432}  - \frac{243}{100352} - \ldots</math>
Other series with the same polynomials include these examples:

<math display="block">\gamma= -\log(a+1) - \sum_{n=1}^\infty\frac{(-1)^n \psi_{n}(a)}{n},\qquad \Re(a)>-1 </math>

and

<math display="block">\gamma= -\frac{2}{1+2a} \left\{\log\Gamma(a+1) -\frac{1}{2}\log(2\pi) + \frac{1}{2} + \sum_{n=1}^\infty\frac{(-1)^n \psi_{n+1}(a)}{n}\right\},\qquad \Re(a)>-1 </math>

where {{math|Γ({{var|a}})}} is the [[gamma function]].{{r|Blagouchine2018}}

A series related to the Akiyama–Tanigawa algorithm is

<math display="block">\gamma= \log(2\pi) - 2 - 2 \sum_{n=1}^\infty\frac{(-1)^n G_{n}(2)}{n}=
\log(2\pi) - 2 + \frac{2}{3} + \frac{1}{24}+ \frac{7}{540} + \frac{17}{2880}+ \frac{41}{12600} + \ldots  </math>

where {{math|{{var|G}}{{sub|{{var|n}}}}(2)}} are the [[Gregory coefficients]] of the second order.{{r|Blagouchine2018}}
  
As a series of [[prime number]]s:

<math display="block">\gamma = \lim_{n\to\infty}\left(\log n - \sum_{p\le n}\frac{\log p}{p-1}\right).</math>

=== Asymptotic expansions ===
{{mvar|γ}} equals the following asymptotic formulas (where {{math|{{var|H}}{{sub|{{var|n}}}}}} is the {{mvar|n}}th [[harmonic number]]):

*<math display="inline">\gamma \sim H_n - \log n - \frac1{2n} + \frac1{12n^2} - \frac1{120n^4} + \cdots</math> (''Euler'')
*<math display="inline">\gamma \sim H_n - \log\left({n + \frac1{2} + \frac1{24n} - \frac1{48n^2} + \cdots}\right)</math> (''Negoi'')
*<math display="inline">\gamma \sim H_n - \frac{\log n + \log(n+1)}{2} - \frac1{6n(n+1)} + \frac1{30n^2(n+1)^2} - \cdots</math> (''[[Ernesto Cesàro|Cesàro]]'')

The third formula is also called the Ramanujan expansion.

Alabdulmohsin derived closed-form expressions for the sums of errors of these approximations.{{r|Alabdulmohsin2018_1478}} He showed that (Theorem A.1):

<math display="block">\begin{align}
\sum_{n=1}^\infty \log n +\gamma - H_n + \frac{1}{2n} &= \frac{\log (2\pi)-1-\gamma}{2} \\
\sum_{n=1}^\infty \log \sqrt{n(n+1)} +\gamma - H_n &= \frac{\log (2\pi)-1}{2}-\gamma \\
\sum_{n=1}^\infty (-1)^n\Big(\log n +\gamma - H_n\Big) &= \frac{\log \pi-\gamma}{2}
\end{align}</math>

=== Exponential ===
The constant {{math|{{var|e}}{{i sup|1px|{{var|γ}}}}}} is important in number theory.  Its numerical value is:{{r|OEIS_A073004}}

{{block indent|{{gaps|1.78107|24179|90197|98523|65041|03107|17954|91696|45214|30343|...}}.}}
{{math|{{var|e}}{{i sup|1px|{{var|γ}}}}}} equals the following [[limit of a sequence|limit]], where {{math|{{var|p}}{{sub|{{var|n}}}}}} is the {{mvar|n}}th [[prime number]]:

<math display="block">e^\gamma = \lim_{n\to\infty}\frac1{\log p_n} \prod_{i=1}^n \frac{p_i}{p_i-1}.</math>

This restates the third of [[Mertens' theorems]].{{r|excursions}}

We further have the following product involving the three constants {{math|{{var|e}}}}, {{math|{{var|π}}}} and {{math|{{var|γ}}}}:<ref name=":9">{{Cite web |last=Weisstein |first=Eric W. |title=Mertens Theorem |url=https://mathworld.wolfram.com/MertensTheorem.html |access-date=2024-10-08 |website=mathworld.wolfram.com |language=en}}</ref>

<math display="block">\frac{\pi^2}{6e^\gamma}=\lim_{n\to\infty}\frac1{\log p_n} \prod_{i=1}^n \frac{p_i}{p_i+1}.</math>

Other [[infinite product]]s relating to {{math|{{var|e}}{{i sup|1px|{{var|γ}}}}}} include:

<math display="block">\begin{align}
\frac{e^{1+\frac{\gamma}{2}}}{\sqrt{2\pi}} &= \prod_{n=1}^\infty e^{-1+\frac1{2n}}\left(1+\frac1{n}\right)^n \\
\frac{e^{3+2\gamma}}{2\pi} &= \prod_{n=1}^\infty e^{-2+\frac2{n}}\left(1+\frac2{n}\right)^n. \end{align}</math>

These products result from the [[Barnes G-function|Barnes {{mvar|G}}-function]].

In addition,

<math display="block">e^{\gamma} = \sqrt{\frac2{1}} \cdot \sqrt[3]{\frac{2^2}{1\cdot 3}} \cdot \sqrt[4]{\frac{2^3\cdot 4}{1\cdot 3^3}} \cdot \sqrt[5]{\frac{2^4\cdot 4^4}{1\cdot 3^6\cdot 5}} \cdots</math>

where the {{mvar|n}}th factor is the {{math|({{var|n}} + 1)}}th root of

<math display="block">\prod_{k=0}^n (k+1)^{(-1)^{k+1}{n \choose k}}.</math>

This infinite product, first discovered by Ser in 1926, was rediscovered by Sondow using [[hypergeometric function]]s.{{r|Sondow2003}}

It also holds that{{r|ChoiSrivastava2010}}

<math display="block">\frac{e^\frac{\pi}{2}+e^{-\frac{\pi}{2}}}{\pi e^\gamma}=\prod_{n=1}^\infty\left(e^{-\frac{1}{n}}\left(1+\frac{1}{n}+\frac{1}{2n^2}\right)\right).</math>

==Published digits==
{| class="wikitable" style="margin: 1em auto 1em auto"
|+ Published decimal expansions of {{mvar|γ}}
! Date || Decimal digits || Author || Sources
|-
| 1734 || style="text-align:right;"| 5 || [[Leonhard Euler]] ||{{sfn|Lagarias|2013}}
|-
| 1735 || style="text-align:right;"| 15 || Leonhard Euler ||{{sfn|Lagarias|2013}}
|-
| 1781 || style="text-align:right;"| 16 || Leonhard Euler ||{{sfn|Lagarias|2013}}
|-
| 1790 || style="text-align:right;"| 32 || [[Lorenzo Mascheroni]], with 20–22 and 31–32 wrong ||{{sfn|Lagarias|2013}}
|-
| 1809 || style="text-align:right;"| 22 || [[Johann Georg von Soldner|Johann G. von Soldner]] ||{{sfn|Lagarias|2013}}
|-
| 1811 || style="text-align:right;"| 22 || [[Carl Friedrich Gauss]] ||{{sfn|Lagarias|2013}}
|-
| 1812 || style="text-align:right;"| 40 || [[Friedrich Bernhard Gottfried Nicolai]] ||{{sfn|Lagarias|2013}}
|-
| 1861 || style="text-align:right;" | 41 || Ludwig Oettinger ||<ref>{{Cite journal |last=Oettinger |first=Ludwig |date=1862-01-01 |title=Ueber die richtige Werthbestimmung der Constante des Integrallogarithmus. |url=https://www.degruyter.com/document/doi/10.1515/crll.1862.60.375/html? |journal=Journal für die reine und angewandte Mathematik |language=de |volume=60 |issue= |pages=375–376 |doi=10.1515/crll.1862.60.375 |issn=1435-5345}}</ref>
|-
| 1867 || style="text-align:right;" | 49 || [[William Shanks]]||<ref>{{Cite journal |date=1867-12-31 |title=I. On the calculation of the numerical value of Euler's constant, which Professor Price, of Oxford, calls E |url=https://royalsocietypublishing.org/doi/10.1098/rspl.1866.0100 |journal=Proceedings of the Royal Society of London |language=en |volume=15 |pages=429–432 |doi=10.1098/rspl.1866.0100 |issn=0370-1662}}</ref>
|-
| 1871 || style="text-align:right;" | 100 || [[James Whitbread Lee Glaisher|James W.L. Glaisher]]||{{sfn|Lagarias|2013}}
|-
| 1877 || style="text-align:right;" | 263 || [[John Couch Adams|J. C. Adams]]||{{sfn|Lagarias|2013}}
|-
| 1952 || style="text-align:right;" | 328 || [[John Wrench|John William Wrench Jr.]]||{{sfn|Lagarias|2013}}
|-
| 1961 || style="text-align:right;" | {{val|1050|fmt=gaps}}|| Helmut Fischer and [[Karl Longin Zeller|Karl Zeller]]|| <ref>{{Cite web |last1=Fischer |first1=Helmut |last2=Zeller |first2=Karl |date=1961 |title=Bernoullische Zahlen und Eulersche Konstante |url=https://zbmath.org/?q=Bernoullische+Zahlen+und+Eulersche+Konstante |access-date=2024-10-27 |website=zbmath.org |language=de}}</ref>
|-
| 1962 || style="text-align:right;"| {{val|1271|fmt=gaps}} || [[Donald Knuth]] ||{{r|Knuth1962}}
|-
| 1963 || style="text-align:right;" | {{val|3566|fmt=gaps}} || Dura W. Sweeney ||<ref>{{Cite journal |last=Sweeney |first=Dura W. |date=1963 |title=On the computation of Euler's constant |journal=Mathematics of Computation |language=en |volume=17 |issue=82 |pages=170–178 |doi=10.1090/S0025-5718-1963-0160308-X |s2cid=120162114 |issn=0025-5718}}</ref>
|-
| 1973 || style="text-align:right;"| {{val|4879|fmt=gaps}} || William A. Beyer and [[Michael S. Waterman]] ||<ref>{{Cite journal |last1=Beyer |first1=W. A. |last2=Waterman |first2=M. S. |date=1974 |title=Error Analysis of a Computation of Euler's Constant |url=https://www.jstor.org/stable/2005935 |journal=Mathematics of Computation |volume=28 |issue=126 |pages=599–604 |doi=10.2307/2005935 |jstor=2005935 |issn=0025-5718}}</ref>
|-
| 1977 || style="text-align:right;"| {{val|20700}} || [[Richard Brent (scientist)|Richard P. Brent]] ||<ref name=":4" />
|-
| 1980 || style="text-align:right;"| {{val|30100}} || Richard P. Brent & [[Edwin McMillan|Edwin M. McMillan]] ||<ref>{{Cite journal |last1=Brent |first1=Richard P. |last2=McMillan |first2=Edwin M. |date=1980 |title=Some new algorithms for high-precision computation of Euler's constant |url=https://www.ams.org/journals/mcom/1980-34-149/S0025-5718-1980-0551307-4/ |journal=Mathematics of Computation |language=en |volume=34 |issue=149 |pages=305–312 |doi=10.1090/S0025-5718-1980-0551307-4 |issn=0025-5718}}</ref>
|-
| 1993 || style="text-align:right;"| {{val|172000}} || [[Jonathan Borwein]] ||<ref name=":5">{{Cite web |last1=Gourdon |first1=Xavier |last2=Sebah |first2=Pascal |date=2004 |title=The Euler constant: γ |url=https://scipp.ucsc.edu/~haber/archives/physics116A10/euler.pdf |access-date=2024-10-27 |website=scipp.ucsc.edu}}</ref>
|-
| 1997 || style="text-align:right;" | {{val|1000000}}|| Thomas Papanikolaou ||<ref name=":5">{{Cite web |last1=Gourdon |first1=Xavier |last2=Sebah |first2=Pascal |date=2004 |title=The Euler constant: γ |url=https://scipp.ucsc.edu/~haber/archives/physics116A10/euler.pdf |access-date=2024-10-27 |website=scipp.ucsc.edu}}</ref>
|-
| 1998 || style="text-align:right;" | {{val|7286255}}|| Xavier Gourdon ||<ref name=":5">{{Cite web |last1=Gourdon |first1=Xavier |last2=Sebah |first2=Pascal |date=2004 |title=The Euler constant: γ |url=https://scipp.ucsc.edu/~haber/archives/physics116A10/euler.pdf |access-date=2024-10-27 |website=scipp.ucsc.edu}}</ref>
|-
| 1999 || style="text-align:right;"| {{val|108000000}} || Patrick Demichel and Xavier Gourdon ||<ref name=":5" />
|-
| March 13, 2009 || style="text-align:right;"| {{val|29844489545}} || Alexander J. Yee & Raymond Chan ||{{r|Yee2011|Y_cruncher}}
|-
| December 22, 2013 || style="text-align:right;"| {{val|119,377,958,182}} || Alexander J. Yee ||{{r|Y_cruncher}}
|-
| March 15, 2016 || style="text-align:right;"| {{val|160,000,000,000}} || Peter Trueb ||{{r|Y_cruncher}}
|-
| May 18, 2016 || style="text-align:right;"| {{val|250,000,000,000}} || Ron Watkins ||{{r|Y_cruncher}}
|-
| August 23, 2017 || style="text-align:right;"| {{val|477,511,832,674}} || Ron Watkins ||{{r|Y_cruncher}}
|-
| May 26, 2020 || style="text-align:right;"| {{val|600,000,000,100}} || Seungmin Kim & Ian Cutress ||{{r|Y_cruncher|MascheroniConst_PolCol}}
|-
| May 13, 2023 || style="text-align:right;"| {{val|700,000,000,000}} || Jordan Ranous & Kevin O'Brien ||{{r|Y_cruncher}}
|-
| September 7, 2023 || style="text-align:right;"| {{val|1,337,000,000,000}} || Andrew Sun || {{r|Y_cruncher}}
|}

==Generalizations==
=== Stieltjes constants ===
{{Main|Stieltjes constants}}
[[File:Generalisation of Euler–Mascheroni constant.jpg|thumb|upright=2|Euler's generalized constants {{math|abm({{var|-<math>\alpha</math>}})}} for {{math|{{var|α}} > 0}}.|alt=]]
''Euler's generalized constants'' are given by

<math display="block">\gamma_\alpha = \lim_{n\to\infty}\left(\sum_{k=1}^n \frac1{k^\alpha} - \int_1^n \frac1{x^\alpha}\,dx\right)</math>

for {{math|0 < {{var|α}} < 1}}, with {{mvar|γ}} as the special case {{math|1={{var|α}} = 1}}.{{sfn|Havil|2003|pp=117–18}}  Extending for {{math| {{var|α}} > 1}} gives:

<math display="block">\gamma_{\alpha} = \zeta(\alpha) - \frac1{\alpha-1}</math>

with again the limit:

<math display="block">\gamma = \lim_{a\to 1}\left(\zeta(a) - \frac1{a-1}\right)</math>

This can be further generalized to

<math display="block">c_f = \lim_{n\to\infty}\left(\sum_{k=1}^n f(k) - \int_1^n f(x)\,dx\right)</math>

for some arbitrary decreasing function {{mvar|f}}. Setting

<math display="block">f_n(x) = \frac{(\log x)^n}{x}</math>

gives rise to the [[Stieltjes constants]] <math>\gamma_n</math>, that occur in the [[Laurent series]] expansion of the [[Riemann zeta function]]:

: <math>\zeta(1+s)=\frac{1}{s}+\sum_{n=0}^\infty \frac{(-1)^n}{n!} \gamma_n s^n.</math>

with <math>\gamma_0 = \gamma = 0.577\dots</math>
{| class="wikitable"
|''n''
|approximate value of γ<sub>''n''</sub>
|[[OEIS]]
|-
|0
| +0.5772156649015
|{{OEIS link|A001620}}
|-
|1
|−0.0728158454836
|{{OEIS link|A082633}}
|-
|2
|−0.0096903631928
|{{OEIS link|A086279}}
|-
|3
| +0.0020538344203
|{{OEIS link|A086280}}
|-
|4
| +0.0023253700654
|{{OEIS link|A086281}}
|-
|100
|−4.2534015717080 × 10<sup>17</sup>
|
|-
|1000
|−1.5709538442047 × 10<sup>486</sup>
|
|}

=== Euler-Lehmer constants ===

''Euler–Lehmer constants'' are given by summation of inverses of numbers in a common
modulo class:{{r|RamMurtySaradha2010}}

<math display="block">\gamma(a,q) = \lim_{x\to \infty}\left (\sum_{0<n\le x \atop n\equiv a \pmod q} \frac1{n}-\frac{\log x}{q}\right).</math>

The basic properties are

<math display="block">\begin{align}
&\gamma(0,q) = \frac{\gamma -\log q}{q}, \\
&\sum_{a=0}^{q-1} \gamma(a,q)=\gamma, \\
&q\gamma(a,q) = \gamma-\sum_{j=1}^{q-1}e^{-\frac{2\pi aij}{q}}\log\left(1-e^{\frac{2\pi ij}{q}}\right),
\end{align}</math>

and if the [[greatest common divisor]] {{math|1=gcd({{var|a}},{{var|q}}) = {{var|d}}}} then

<math display="block">q\gamma(a,q) = \frac{q}{d}\gamma\left(\frac{a}{d},\frac{q}{d}\right)-\log d.</math>

===Masser-Gramain constant===

A two-dimensional generalization of Euler's constant is the [[Masser-Gramain constant]]. It is defined as the following limiting difference:<ref>{{Cite web |last=Weisstein |first=Eric W. |title=Masser-Gramain Constant |url=https://mathworld.wolfram.com/Masser-GramainConstant.html |access-date=2024-10-19 |website=mathworld.wolfram.com |language=en}}</ref>

:<math>\delta = \lim_{n\to\infty} \left( -\log n + \sum_{k=2}^n \frac{1}{\pi r_k^2} \right)</math>

where <math>r_k</math> is the smallest radius of a disk in the complex plane containing at least <math>k</math> [[Gaussian integer|Gaussian integers]].

The following bounds have been established: <math>1.819776 < \delta < 1.819833</math>.<ref>{{Cite web |last1=Melquiond |first1=Guillaume |last2=Nowak |first2=W. Georg |last3=Zimmermann |first3=Paul |title=Numerical approximation of the Masser-Gramain constant to four decimal digits |url=https://www.lri.fr/~melquion/doc/12-mc.pdf |access-date=2024-10-03}}</ref>

== See also ==
* [[Harmonic series (mathematics)|Harmonic series]]
* [[Riemann zeta function]]
* [[Stieltjes constants]]
* [[Meissel-Mertens constant]]
== References ==
* {{cite journal |last=Bretschneider |first=Carl Anton |date=1837 |orig-date=1835 |title=Theoriae logarithmi integralis lineamenta nova |journal=Crelle's Journal |volume=17 |pages=257–285 |language=la |url=https://zenodo.org/record/1448830
}}
* {{cite book |last=Havil |first=Julian |date=2003 |title=Gamma: Exploring Euler's Constant |publisher=Princeton University Press |isbn=978-0-691-09983-5
}}
* {{cite journal |last=Lagarias |first=Jeffrey C. |date=2013 |title=Euler's constant: Euler's work and modern developments |journal=[[Bulletin of the American Mathematical Society]] |volume=50 |issue=4 |page=556 |doi=10.1090/s0273-0979-2013-01423-x |arxiv=1303.1856 |s2cid=119612431
}}
===Footnotes===
{{Reflist|refs=
<ref name=OEIS_A002852>{{Cite OEIS |A002852 |Continued fraction for Euler's constant}}</ref>

<ref name=DeMorgan183642>{{cite book |last=De Morgan |first=Augustus |author-link=Augustus De Morgan |date=1836–1842 |title=The differential and integral calculus |location=London |publisher=Baldwin and Craddoc |at="{{math|''γ''}}" on [https://books.google.com/books?id=95x4IrIcHrgC&pg=PA578 p. 578] }}</ref>

<ref name=CavesFuchs1996>{{cite book |last1=Caves |first1=Carlton M. |author-link1=Carlton M. Caves |last2=Fuchs |first2=Christopher A. |date=1996 |chapter=Quantum information: How much information in a state vector? |title=The Dilemma of Einstein, Podolsky and Rosen – 60 Years Later |publisher=Israel Physical Society |isbn=9780750303941 |oclc=36922834 |arxiv=quant-ph/9601025 |bibcode=1996quant.ph..1025C}}</ref>

<ref name=HaiblePapanikolaou1998>{{Cite book |last1=Haible |first1=Bruno |last2=Papanikolaou |first2=Thomas |title=Algorithmic Number Theory |chapter=Fast multiprecision evaluation of series of rational numbers |date=1998 |editor-last=Buhler |editor-first=Joe P. |series=Lecture Notes in Computer Science |volume=1423 |publisher=Springer |pages=338–350 |doi=10.1007/bfb0054873 |isbn=9783540691136}}</ref>

<ref name=Papanikolaou1997>{{cite thesis |last=Papanikolaou |first=T. |date=1997 |title=Entwurf und Entwicklung einer objektorientierten Bibliothek für algorithmische Zahlentheorie |publisher=Universität des Saarlandes |url=https://www-old.cdc.informatik.tu-darmstadt.de/reports/reports/papa.diss.ps.gz |language=de}}</ref>

<ref name=Sondow2003a>See also {{cite journal |last1=Sondow |first1=Jonathan |date=2003 |title=Criteria for irrationality of Euler's constant |journal=Proceedings of the American Mathematical Society |volume=131 |issue=11 |pages=3335–3344 |doi=10.1090/S0002-9939-03-07081-3 |s2cid=91176597 |arxiv=math.NT/0209070 }}</ref>

<ref name=Mahler1968>{{Cite journal |last1=Mahler |first1=Kurt |last2=Mordell |first2=Louis Joel |date=4 June 1968 |title=Applications of a theorem by A. B. Shidlovski |journal=Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences |volume=305 |issue=1481 |pages=149–173 |doi=10.1098/rspa.1968.0111 |bibcode=1968RSPSA.305..149M |s2cid=123486171}}</ref>

<ref name=RamMurtySaradha2010>{{Cite journal |last1=Ram Murty |first1=M. |first2=N. |last2=Saradha |date=2010 |title=Euler–Lehmer constants and a conjecture of Erdos |journal=Journal of Number Theory |volume=130 |issue=12 |pages=2671–2681 |doi=10.1016/j.jnt.2010.07.004 |issn=0022-314X |doi-access=free }}</ref>

<ref name=Aptekarev2009>{{cite arXiv |last=Aptekarev |first=A. I. |date=28 February 2009 |title=On linear forms containing the Euler constant |class=math.NT |eprint=0902.1768}}</ref>

<ref name=Rivoal2012>{{Cite journal |last=Rivoal |first=Tanguy |date=2012 |title=On the arithmetic nature of the values of the gamma function, Euler's constant, and Gompertz's constant |journal=Michigan Mathematical Journal |volume=61 |issue=2 |pages=239–254 |doi=10.1307/mmj/1339011525 |issn=0026-2285 |doi-access=free |url=https://projecteuclid.org/euclid.mmj/1339011525}}</ref>

<ref name=MurtyZaytseva2013>{{Cite journal |last1=Murty |first1=M. Ram |last2=Zaytseva |first2=Anastasia |date=2013 |title=Transcendence of Generalized Euler Constants |journal=The American Mathematical Monthly |volume=120 |issue=1 |pages=48–54 |doi=10.4169/amer.math.monthly.120.01.048 |jstor=10.4169/amer.math.monthly.120.01.048 |s2cid=20495981 |issn=0002-9890 |url=https://www.jstor.org/stable/10.4169/amer.math.monthly.120.01.048 }}</ref>

<ref name="Krämer2005">{{cite book |last1=Krämer |first1= Stefan |date=2005 |title=Die Eulersche Konstante {{math|<i>γ</i>}} und verwandte Zahlen |publisher=University of Göttingen |language=de }}</ref>

<ref name=Sondow1998>{{cite journal |last1=Sondow |first1=Jonathan |date=1998  |title=An antisymmetric formula for Euler's constant |journal=Mathematics Magazine |volume=71 |issue=3 |pages=219–220 |doi=10.1080/0025570X.1998.11996638 |access-date=2006-05-29 |url=http://home.earthlink.net/~jsondow/id8.html |archive-date=2011-06-04 |archive-url=https://web.archive.org/web/20110604123534/http://home.earthlink.net/~jsondow/id8.html}}</ref>

<ref name=Sondow2005>{{cite journal |last=Sondow |first=Jonathan |date=2005 |title=Double integrals for Euler's constant and <math>\log \frac{4}{\pi}</math> and an analog of Hadjicostas's formula |journal=[[American Mathematical Monthly]] |volume=112 |issue=1 |pages=61–65 |doi=10.2307/30037385 |jstor=30037385 |arxiv=math.CA/0211148}}</ref>

<ref name=Sondow2005a>{{cite book |last1= Sondow |first1=Jonathan |date=1 August 2005a |title=New Vacca-type rational series for Euler's constant and its 'alternating' analog <math>\log \frac{4}{\pi}</math> |arxiv=math.NT/0508042}}</ref>

<ref name=SondowZudilin2006>{{cite journal |last1=Sondow |first1=Jonathan |first2=Wadim |last2=Zudilin |date=2006 |title=Euler's constant, {{math|''q''}}-logarithms, and formulas of Ramanujan and Gosper |journal=The Ramanujan Journal |volume=12 |issue=2 |pages=225–244 |doi=10.1007/s11139-006-0075-1 |s2cid=1368088 |arxiv=math.NT/0304021}}</ref>

<ref name=DeTemple1993>{{cite journal |last=DeTemple |first=Duane W. |date=May 1993 |title=A Quicker Convergence to Euler's Constant |journal=The American Mathematical Monthly |volume=100 |issue=5 |pages=468–470 |doi=10.2307/2324300 |issn=0002-9890 |jstor=2324300}}</ref>

<ref name=Blagouchine2016>{{cite journal |last=Blagouchine |first=Iaroslav V. |date=2016 |title=Expansions of generalized Euler's constants into the series of polynomials in {{math|π<sup>−2</sup>}} and into the formal enveloping series with rational coefficients only |journal=[[Journal of Number Theory|J. Number Theory]] |volume=158 |pages=365–396 |doi=10.1016/j.jnt.2015.06.012 |arxiv=1501.00740}}</ref>

<ref name=Vacca1910>{{cite journal |last1=Vacca |first1=G. |author-link=Giovanni Vacca (mathematician) |date=1910 |title=A new analytical expression for the number π and some historical considerations |journal=Bulletin of the American Mathematical Society |volume=16 |pages=368–369|doi=10.1090/S0002-9904-1910-01919-4 |doi-access=free }}</ref>

<ref name=Vacca1926>{{cite journal |last1=Vacca |first1=G. |author-link=Giovanni Vacca (mathematician) |date=1926 |title=Nuova serie per la costante di Eulero, {{math|''C''}} = 0,577...". Rendiconti, Accademia Nazionale dei Lincei, Roma, Classe di Scienze Fisiche |journal=Matematiche e Naturali |volume=6 |issue=3 |pages=19–20 |language=it}}</ref>

<ref name=Glaisher1910>{{cite journal |last=Glaisher |first=James Whitbread Lee |author-link=James Whitbread Lee Glaisher |date=1910 |title=On Dr. Vacca's series for {{math|''γ''}} |journal=Q. J. Pure Appl. Math. |volume=41 |pages=365–368}}</ref>

<ref name=Hardy1912>{{ cite journal |last1= Hardy |first1=G.H. |date=1912 |title=Note on Dr. Vacca's series for {{math|''γ''}} |journal=Q. J. Pure Appl. Math. |volume=43 |pages=215–216}}</ref>

<ref name=Kluyver1927>{{cite journal |last1=Kluyver |first1=J.C. |date=1927 |title=On certain series of Mr. Hardy |journal=Q. J. Pure Appl. Math. |volume=50 |pages=185–192}}</ref>

<ref name=Blagouchine2018>{{cite journal |last1=Blagouchine |first1=Iaroslav V. |date=2018 |title=Three notes on Ser's and Hasse's representations for the zeta-functions |journal=INTEGERS: The Electronic Journal of Combinatorial Number Theory |volume=18A |issue=#A3 |pages=1–45 |bibcode=2016arXiv160602044B |arxiv=1606.02044 |url=http://math.colgate.edu/~integers/vol18a.html}}</ref>

<ref name=Alabdulmohsin2018_1478>{{cite book |last=Alabdulmohsin |first=Ibrahim M. |date=2018 |title=Summability Calculus. A Comprehensive Theory of Fractional Finite Sums |publisher=[[Springer Science+Business Media|Springer]] |pages=147–8 |isbn=9783319746487}}</ref>

<ref name=OEIS_A302120>{{cite OEIS|A302120|Absolute value of the numerators of a series converging to Euler's constant}}</ref>

<ref name=OEISA302121>{{cite OEIS|A302121|Denominators of a series converging to Euler's constant}}</ref>

<ref name=excursions>{{cite book
 | last = Ramaré | first = Olivier
 | doi = 10.1007/978-3-030-73169-4
 | isbn = 978-3-030-73168-7
 | location = Basel
 | mr = 4400952
 | page = 131
 | publisher = Birkhäuser/Springer
 | series = Birkhäuser Advanced Texts: Basel Textbooks
 | title = Excursions in Multiplicative Number Theory
 | url = https://books.google.com/books?id=n1piEAAAQBAJ&pg=PA131
 | year = 2022| s2cid = 247271545
 }}</ref>

<ref name=OEIS_A073004>{{cite OEIS|A073004|Decimal expansion of exp(gamma)}}</ref>

<ref name=ChoiSrivastava2010>{{Cite journal |last1=Choi |first1=Junesang |last2=Srivastava |first2=H.M. |date=1 September 2010 |title=Integral Representations for the Euler–Mascheroni Constant {{math|''γ''}} |journal=Integral Transforms and Special Functions |volume=21 |issue=9 |pages=675–690 |doi=10.1080/10652461003593294 |s2cid=123698377 |issn=1065-2469}}</ref>

<ref name=Sondow2003>{{cite arXiv |last1=Sondow |first1=Jonathan |date=2003 |eprint=math.CA/0306008 |title= An infinite product for {{math|''e{{sup|γ}}''}} via hypergeometric formulas for Euler's constant, {{math|''γ''}}}}</ref>

<ref name=Knuth1962>{{cite journal |last1=Knuth |first1=Donald E. |author-link=Donald Knuth |date=July 1962 |title=Euler's Constant to 1271 Places |journal=Mathematics of Computation |volume=16 |issue=79 |pages=275–281 |publisher=[[American Mathematical Society]] |doi=10.2307/2004048 |jstor=2004048 |url=https://www.jstor.org/stable/2004048}}</ref>

<ref name=Yee2011>{{cite web |last=Yee |first=Alexander J. |date=7 March 2011 |title=Large Computations |website=www.numberworld.org |url=http://www.numberworld.org/nagisa_runs/computations.html}}</ref>

<ref name=Y_cruncher>{{cite web |last=Yee |first=Alexander J. |title=Records Set by y-cruncher |website=www.numberworld.org |access-date=30 April 2018 |url=http://www.numberworld.org/y-cruncher/records.html }}<br/>
{{cite web |last=Yee |first=Alexander J. |title=y-cruncher - A Multi-Threaded Pi-Program |website=www.numberworld.org |url=http://www.numberworld.org/y-cruncher/}}</ref>

<ref name=MascheroniConst_PolCol>{{cite web |title=Euler-Mascheroni Constant |website=Polymath Collector |date=15 February 2020 |url=https://ehfd.github.io/world-record/euler-mascheroni-constant/}}</ref>

<ref name=ConnallonHodgins2021>{{cite journal
 | last1 = Connallon | first1 = Tim
 | last2 = Hodgins | first2 = Kathryn A.
 | date = October 2021
 | doi = 10.1111/evo.14372
 | issue = 11
 | journal = Evolution
 | pages = 2624–2640
 | title = Allen Orr and the genetics of adaptation
 | volume = 75| pmid = 34606622
 | s2cid = 238357410
 }}</ref>

<ref name=Berndt2008>{{cite journal |last1=Berndt |first1=Bruce C.|author-link=Bruce C. Berndt |date=January 2008 |title=A fragment on Euler's constant in Ramanujan's lost notebook |journal=South East Asian J. Math. & Math. Sc |volume=6 |issue=2 |pages=17–22 |doi= |jstor= |url=https://scholarworks.utrgv.edu/cgi/viewcontent.cgi?article=1130&context=mss_fac}}</ref>

<ref name=A001620>{{Cite OEIS|A001620|Decimal expansion of Euler's constant (or the Euler-Mascheroni constant), gamma}}</ref>

}}

== Further reading ==
* {{cite journal |author1=Borwein, Jonathan M. |author2=David M. Bradley |author3=Richard E. Crandall |title=Computational Strategies for the Riemann Zeta Function |journal=Journal of Computational and Applied Mathematics |date=2000 |volume=121 |issue=1–2 |pages=11 |doi=10.1016/s0377-0427(00)00336-8 |bibcode=2000JCoAM.121..247B |ref=none |doi-access=free }} Derives {{math|''γ''}} as sums over Riemann zeta functions.
* {{cite book |last=Finch |first=Steven R. |date=2003 |title=Mathematical Constants|series= Encyclopedia of Mathematics and its Applications|volume=94|publisher=Cambridge University Press|location=Cambridge|isbn=0-521-81805-2|ref=none}}
* {{cite journal |first1=I. |last1=Gerst |title=Some series for Euler's constant |date=1969 |journal=Amer. Math. Monthly |doi=10.2307/2316370 |volume=76 |issue=3 |pages=237–275 |jstor=2316370 |ref=none}}
* {{cite journal |last=Glaisher |first=James Whitbread Lee |author-link=James Whitbread Lee Glaisher |date=1872 |title=On the history of Euler's constant |journal=Messenger of Mathematics |volume=1 |pages=25–30 |jfm=03.0130.01 |ref=none}}
* {{cite web |last1=Gourdon|first1=Xavier|last2=Sebah|first2=P. |date=2002 |title=Collection of formulae for Euler's constant, {{math|''γ''}} |url=http://numbers.computation.free.fr/Constants/Gamma/gammaFormulas.html |ref=none}}
* {{cite web |last1=Gourdon|first1=Xavier|last2=Sebah|first2=P. |date=2004 |title=The Euler constant: {{math|''γ''}} |url=http://numbers.computation.free.fr/Constants/Gamma/gamma.html |ref=none}}
* {{cite journal |first1=E. A. |last1= Karatsuba |title=Fast evaluation of transcendental functions |journal=Probl. Inf. Transm. |volume =27 |number=44 |pages=339–360 |date=1991 |ref=none}}
* {{cite journal |last1=Karatsuba |first1=E.A. |date=2000 |title=On the computation of the Euler constant {{math|''γ''}} |journal=Journal of Numerical Algorithms |volume=24 |issue=1–2 |pages=83–97 |doi=10.1023/A:1019137125281 |s2cid=21545868 |ref=none}}
* {{cite book |last=Knuth |first=Donald |author-link=Donald Knuth |date=1997 |title=[[The Art of Computer Programming|The Art of Computer Programming, Vol. 1]] |edition=3rd |publisher=Addison-Wesley |isbn=0-201-89683-4 |pages=75, 107, 114, 619–620 |ref=none}}
* {{cite journal |first1=D. H. |last1=Lehmer |date=1975 |title=Euler constants for arithmetical progressions |journal=Acta Arith. |volume=27 |number=1 |pages=125–142 |url=http://matwbn.icm.edu.pl/ksiazki/aa/aa27/aa27121.pdf |doi=10.4064/aa-27-1-125-142 |ref=none|doi-access=free }}
* {{cite journal |last1=Lerch |first1=M. |date=1897 |title=Expressions nouvelles de la constante d'Euler |journal=Sitzungsberichte der Königlich Böhmischen Gesellschaft der Wissenschaften |volume=42 |page=5 |ref=none}}
* {{cite book |last=Mascheroni |first=Lorenzo |author-link=Lorenzo Mascheroni |date=1790 |title=Adnotationes ad calculum integralem Euleri, in quibus nonnulla problemata ab Eulero proposita resolvuntur |publisher=Galeati, Ticini |ref=none}}
* {{cite journal |last1=Sondow |first1=Jonathan |date=2002 |title=A hypergeometric approach, via linear forms involving logarithms, to irrationality criteria for Euler's constant |arxiv=math.NT/0211075 |journal=Mathematica Slovaca |volume=59 |pages=307–314 |doi=10.2478/s12175-009-0127-2 |bibcode=2002math.....11075S |s2cid=16340929 |ref=none}} with an Appendix by [https://web.archive.org/web/20130523085959/http://wain.mi.ras.ru/zlobin/ Sergey Zlobin]

== External links ==
* {{springer|title=Euler constant|id=p/e036420|mode=cs1}}
* {{MathWorld|urlname=Euler-MascheroniConstant|title=Euler–Mascheroni constant|ref=none}}
* [https://jonathansondow.github.io/ Jonathan Sondow.]
* [http://www.ccas.ru/personal/karatsuba/algen.htm Fast Algorithms and the FEE Method], E.A. Karatsuba (2005)
* Further formulae which make use of the constant: [http://numbers.computation.free.fr/Constants/Gamma/gammaFormulas.html Gourdon and Sebah (2004).]

{{Leonhard Euler}}
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{{DEFAULTSORT:Euler's constant}}
[[Category:Mathematical constants]]
[[Category:Unsolved problems in number theory]]
[[Category:Leonhard Euler]]