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[{"@context":"http:\/\/schema.org\/","@type":"BlogPosting","@id":"https:\/\/wiki.edu.vn\/en\/wiki3\/dirichlet-beta-function-wikipedia\/#BlogPosting","mainEntityOfPage":"https:\/\/wiki.edu.vn\/en\/wiki3\/dirichlet-beta-function-wikipedia\/","headline":"Dirichlet beta function – Wikipedia","name":"Dirichlet beta function – Wikipedia","description":"From Wikipedia, the free encyclopedia The Dirichlet beta function In mathematics, the Dirichlet beta function (also known as the Catalan","datePublished":"2016-10-28","dateModified":"2016-10-28","author":{"@type":"Person","@id":"https:\/\/wiki.edu.vn\/en\/wiki3\/author\/lordneo\/#Person","name":"lordneo","url":"https:\/\/wiki.edu.vn\/en\/wiki3\/author\/lordneo\/","image":{"@type":"ImageObject","@id":"https:\/\/secure.gravatar.com\/avatar\/c9645c498c9701c88b89b8537773dd7c?s=96&d=mm&r=g","url":"https:\/\/secure.gravatar.com\/avatar\/c9645c498c9701c88b89b8537773dd7c?s=96&d=mm&r=g","height":96,"width":96}},"publisher":{"@type":"Organization","name":"Enzyklop\u00e4die","logo":{"@type":"ImageObject","@id":"https:\/\/wiki.edu.vn\/wiki4\/wp-content\/uploads\/2023\/11\/book.png","url":"https:\/\/wiki.edu.vn\/wiki4\/wp-content\/uploads\/2023\/11\/book.png","width":600,"height":60}},"image":{"@type":"ImageObject","@id":"https:\/\/upload.wikimedia.org\/wikipedia\/commons\/thumb\/e\/e3\/Mplwp_dirichlet_beta.svg\/325px-Mplwp_dirichlet_beta.svg.png","url":"https:\/\/upload.wikimedia.org\/wikipedia\/commons\/thumb\/e\/e3\/Mplwp_dirichlet_beta.svg\/325px-Mplwp_dirichlet_beta.svg.png","height":"217","width":"325"},"url":"https:\/\/wiki.edu.vn\/en\/wiki3\/dirichlet-beta-function-wikipedia\/","wordCount":5793,"articleBody":"From Wikipedia, the free encyclopedia The Dirichlet beta functionIn mathematics, the Dirichlet beta function (also known as the Catalan beta function) is a special function, closely related to the Riemann zeta function. It is a particular Dirichlet L-function, the L-function for the alternating character of period four.Definition[edit]The Dirichlet beta function is defined as\u03b2(s)=\u2211n=0\u221e(\u22121)n(2n+1)s,{displaystyle beta (s)=sum _{n=0}^{infty }{frac {(-1)^{n}}{(2n+1)^{s}}},}or, equivalently,\u03b2(s)=1\u0393(s)\u222b0\u221exs\u22121e\u2212x1+e\u22122xdx.{displaystyle beta (s)={frac {1}{Gamma (s)}}int _{0}^{infty }{frac {x^{s-1}e^{-x}}{1+e^{-2x}}},dx.}In each case, it is assumed that Re(s)\u00a0>\u00a00.Alternatively, the following definition, in terms of the Hurwitz zeta function, is valid in the whole complex s-plane:[1]\u03b2(s)=4\u2212s(\u03b6(s,14)\u2212\u03b6(s,34)).{displaystyle beta (s)=4^{-s}left(zeta left(s,{1 over 4}right)-zeta left(s,{3 over 4}right)right).}Another equivalent definition, in terms of the Lerch transcendent, is:\u03b2(s)=2\u2212s\u03a6(\u22121,s,12),{displaystyle beta (s)=2^{-s}Phi left(-1,s,{{1} over {2}}right),}which is once again valid for all complex values of s.The Dirichlet beta function can also be written in terms of the Polylogarithm function:\u03b2(s)=i2(Lis(\u2212i)\u2212Lis(i)).{displaystyle beta (s)={frac {i}{2}}left({text{Li}}_{s}(-i)-{text{Li}}_{s}(i)right).}Also the series representation of Dirichlet beta function can be formed in terms of the polygamma function\u03b2(s)=12s\u2211n=0\u221e(\u22121)n(n+12)s=1(\u22124)s(s\u22121)![\u03c8(s\u22121)(14)\u2212\u03c8(s\u22121)(34)]{displaystyle beta (s)={frac {1}{2^{s}}}sum _{n=0}^{infty }{frac {(-1)^{n}}{left(n+{frac {1}{2}}right)^{s}}}={frac {1}{(-4)^{s}(s-1)!}}left[psi ^{(s-1)}left({frac {1}{4}}right)-psi ^{(s-1)}left({frac {3}{4}}right)right]}but this formula is only valid at positive integer values of s{displaystyle s}.Euler product formula[edit]It is also the simplest example of a series non-directly related to \u03b6(s){displaystyle zeta (s)} which can also be factorized as an Euler product, thus leading to the idea of Dirichlet character defining the exact set of Dirichlet series having a factorization over the prime numbers.At least for Re(s)\u00a0\u2265\u00a01:\u03b2(s)=\u220fp\u22611\u00a0mod\u00a0411\u2212p\u2212s\u220fp\u22613\u00a0mod\u00a0411+p\u2212s{displaystyle beta (s)=prod _{pequiv 1 mathrm {mod} 4}{frac {1}{1-p^{-s}}}prod _{pequiv 3 mathrm {mod} 4}{frac {1}{1+p^{-s}}}}where p\u22611 mod 4 are the primes of the form 4n+1 (5,13,17,…) and p\u22613 mod 4 are the primes of the form 4n+3 (3,7,11,…). This can be written compactly as"},{"@context":"http:\/\/schema.org\/","@type":"BreadcrumbList","itemListElement":[{"@type":"ListItem","position":1,"item":{"@id":"https:\/\/wiki.edu.vn\/en\/wiki3\/#breadcrumbitem","name":"Enzyklop\u00e4die"}},{"@type":"ListItem","position":2,"item":{"@id":"https:\/\/wiki.edu.vn\/en\/wiki3\/dirichlet-beta-function-wikipedia\/#breadcrumbitem","name":"Dirichlet beta function – Wikipedia"}}]}]