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

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

where

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

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

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

Formulae

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

where $E_n$ are the Euler numbers.

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

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

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

See also

• Catalan's constant ($\beta (2)$)

External links

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