Editing Catalan's constant (section)

==Properties==

{{unsolved|mathematics|Is Catalan's constant irrational? If so, is it transcendental?}}
It is not known whether {{mvar|G}} is [[irrational number|irrational]], let alone [[transcendental number|transcendental]].<ref>{{citation
 | last = Nesterenko | first = Yu. V.
 | date = January 2016
 | doi = 10.1134/s0081543816010107
 | issue = 1
 | journal = Proceedings of the Steklov Institute of Mathematics
 | pages = 153–170
 | title = On Catalan's constant
 | volume = 292| s2cid = 124903059
 }}.</ref> {{mvar|G}} has been called "arguably the most basic constant whose irrationality and transcendence (though strongly
suspected) remain unproven".<ref>{{citation
 | last1 = Bailey | first1 = David H.
 | last2 = Borwein | first2 = Jonathan M.
 | last3 = Mattingly | first3 = Andrew
 | last4 = Wightwick | first4 = Glenn
 | doi = 10.1090/noti1015
 | issue = 7
 | journal = [[Notices of the American Mathematical Society]]
 | mr = 3086394
 | pages = 844–854
 | title = The computation of previously inaccessible digits of <math>\pi^2</math> and Catalan's constant
 | volume = 60
 | year = 2013| doi-access = free
 }}</ref>

There exist however partial results. It is known that infinitely many of the numbers ''β''(2''n'') are irrational, where ''β(s)'' is the Dirichlet beta function.<ref>{{Cite journal |last1=Rivoal |first1=T. |last2=Zudilin |first2=W. |date=2003-08-01 |title=Diophantine properties of numbers related to Catalan's constant |url=https://link.springer.com/article/10.1007/s00208-003-0420-2 |journal=Mathematische Annalen |language=en |volume=326 |issue=4 |pages=705–721 |doi=10.1007/s00208-003-0420-2 |hdl=1959.13/803688 |issn=1432-1807}}</ref> In particular at least one of ''β''(2), ''β''(4), ''β''(6), ''β''(8), ''β''(10) and ''β''(12) must be irrational, where ''β''(2) is Catalan's constant.<ref>{{Cite journal |last=Zudilin |first=Wadim |date=2018-04-26 |title=Arithmetic of Catalan's constant and its relatives |journal=Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg |volume=89 |pages=45–53 |doi=10.1007/s12188-019-00203-w |arxiv=1804.09922 |language=en}}</ref> These results by [[Wadim Zudilin]] and [[Tanguy Rivoal]] are related to similar ones given for the [[Particular values of the Riemann zeta function#Odd positive integers|odd zeta constants]] ζ(2''n+1'').

Catalan's constant is known to be an [[Period (algebraic geometry)|algebraic period]], which follows from some of the double integrals given below.