Dirichlet Beta Function in terms of Hurwitz Zeta Function

Theorem

$\map \beta s = \dfrac 1 {4^s} \paren {\map \zeta {s, \dfrac 1 4} - \map \zeta {s, \dfrac 3 4} }$

where:

$\map \beta s$ is the Dirichlet beta function

$\map \zeta {s, x}$ is the Hurwitz zeta function

$s$ is a complex number with $\map \Re s > 1$

Proof

\(\ds \map \beta s\)

\(=\)

\(\ds \sum_{n \mathop = 0}^\infty \frac {\paren {-1}^n} {\paren {2 n + 1}^s}\)

Definition of Dirichlet Beta Function

\(\ds \)

\(=\)

\(\ds \sum_{n \mathop = 0}^\infty \frac 1 {\paren {4 n + 1}^s} - \sum_{n \mathop = 0}^\infty \frac 1 {\paren {4 n + 3}^s}\)

splitting summation into positive and negative parts

\(\ds \)

\(=\)

\(\ds \frac 1 {4^s} \paren {\sum_{n \mathop = 0}^\infty \frac 1 {\paren {n + \frac 1 4 }^s} - \sum_{n \mathop = 0}^\infty \frac 1 {\paren {n + \frac 3 4}^s} }\)

factoring out $\dfrac 1 {4^s}$

\(\ds \)

\(=\)

\(\ds \frac 1 {4^s} \paren {\map \zeta {s, \frac 1 4} - \map \zeta {s, \frac 3 4} }\)

Definition of Hurwitz Zeta Function

Our reliance upon the Hurwitz zeta function requires that $s \in \C$ and $\map \Re s > 1$.

$\blacksquare$