Editing Euler's constant (section)

== 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 display=inline>\frac \pi 2\frac{Y_0(2)}{J_0(2)}-\gamma</math> is transcendental, where <math>J_0</math> and <math>Y_0</math> are the usual [[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, i. e. 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, if {{math|''q'' ≥ 2}} and {{math|1 ≤ ''a'' < ''q''}}; this family includes the special case {{math|1=''γ''(2,4) = ''γ''/4}}.{{sfn|Lagarias|2013}}{{r|RamMurtySaradha2010}} 

Using the same approach, in 2013, M. Ram Murty and A. Zaytseva showed that the generalized Euler constants have the same property,
{{sfn|Lagarias|2013}}{{r|MurtyZaytseva2013}}
<ref>{{Cite journal |last1=Diamond |first1=H. G. |last2=Ford |first2=K. |title=Generalized Euler constants |journal=Mathematical Proceedings of the Cambridge Philosophical Society |date=2008 |volume=145 |issue=1 |pages=27–41 |publisher=[[Cambridge University Press]] |doi=10.1017/S0305004108001187 |arxiv=math/0703508|bibcode=2008MPCPS.145...27D }}</ref> where the generalized Euler constant are defined as
<math display="block">\gamma(\Omega) = \lim_{x\rightarrow\infty} \left( \sum_{n=1}^x \frac{1_\Omega(n)}{n} - \log x \cdot \lim_{x\rightarrow\infty} \frac{ \sum_{n=1}^x 1_\Omega (n) }{x} \right),</math>
where {{tmath|\Omega}} is a fixed list of prime numbers, <math>1_\Omega(n) =0</math> if at least one of the primes in {{tmath|\Omega}} is a prime factor of {{tmath|n}}, and <math>1_\Omega(n) =1</math> otherwise. In particular, {{tmath|1=\gamma(\empty)=\gamma}}.

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.