Dirichlet beta function

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The Dirichlet beta function (also known as the Catalan beta function) is a special function closely related to the Riemann zeta function. It is defined as

$\beta (s):=\sum _{n=0}^{\infty }{\frac {(-1)^{n}}{(2n+1)^{s}}}=\sum _{n=1}^{\infty }{\frac {{\chi _{\beta }}(n)}{n^{s}}},\quad s>0,\,$

where

${\chi _{\beta }}(n)=\cos \,{\frac {(n-1)\pi }{2}},\quad n\geq 1,\,$

is the alternating character (of period four) of the above Dirichlet L-function, giving the sequence (A056594)

{1, 0, -1, 0, 1, 0, -1, 0, 1, 0, -1, 0, 1, 0, -1, 0, 1, 0, -1, 0, 1, 0, -1, 0, 1, 0, -1, 0, 1, 0, -1, 0, 1, 0, -1, 0, 1, 0, -1, 0, 1, 0, -1, 0, 1, 0, -1, 0, 1, 0, -1, 0, 1, 0, -1, 0, ...}

with generating function

$G_{\{{\chi _{\scriptscriptstyle \beta }}(n)\}}(x)={\frac {x}{1+x^{2}}},\quad n\geq 1.\,$

Formulae

$\beta (2n+1)=(-1)^{n}\,{\frac {E_{2n}}{2\,(2n)!}}\left({\frac {\pi }{2}}\right)^{2n+1},\quad n\geq 0,\,$

where $\scriptstyle E_{n}\,$ are the Euler numbers.

$\beta (2n)={\frac {\psi ^{(2n-1)}({\frac {1}{4}})}{2\,(2n-1)!\,4^{2n-1}}}-{\frac {(2^{2n-1}-{\frac {1}{2}})\,|B_{2n}|\,\pi ^{2n}}{(2n)!}}={\frac {\psi ^{(2n-1)}({\frac {1}{4}})}{2\,(2n-1)!\,4^{2n-1}}}-(1-2^{-2n})\,\zeta (2n),\quad n\geq 1,\,$

where $\scriptstyle \psi ^{(n)}(x)\,$ is the polygamma function, $\scriptstyle B_{n}\,$ are the Bernoulli numbers and the Riemann zeta function for even integers is given by

$\zeta (2n)={\frac {2^{2n-1}\,|B_{2n}|\,\pi ^{2n}}{(2n)!}},\quad n\geq 1.\,$

See also

• Catalan's constant ($\scriptstyle \beta (2)\,$)

External links

• Weisstein, Eric W., Dirichlet Beta Function, from MathWorld—A Wolfram Web Resource.