As a Dirichlet series
The Riemann zeta-function is the following Dirichlet series of :
It can also be expressed as a product, using the product formula for Dirichlet series of completely multiplicative function:
As the function obtained by analytic continuation
The Dirichlet series for the Riemann zeta-function is absolutely convergent for all complex numbers for which . Although the series does not make sense for other , the function extends to a meromorphic function of , with a single simple pole at the point .
In terms of the Dirichlet eta-function
The Riemann zeta-function can be defined in terms of the Dirichlet eta-function:
The Dirichlet eta-function is also defined in terms of a Dirichlet series, but this Dirichlet series has the advantage of being convergent on a wider range.
Similarly defined functions
Zeros and poles
The Riemann zeta-function has a single simple pole at .
The Riemann zeta-function has zeros at all negative even integers. There are also infinitely many zeros with real part . The Riemann hypothesis states that all zeros other than the negative even integers have real part .