Dirichlet eta function
The Dirichlet eta function can be defined as
where ζ is Riemann's zeta function. However, it can also be used to define the zeta function. It has a Dirichlet series expression, valid for any complex number s with positive real part, given by
While this is convergent only for s with positive real part, it is Abel summable for any complex number, which serves to define the eta function as an entire function, and shows the zeta function is meromorphic with a single pole at .
Hardy gave a simple proof of the functional equation for the eta function, which is
From this, one immediately has the functional equation of the zeta function also.
Also see
