Riemann zeta-function

Definition

As a Dirichlet series

The Riemann zeta-function is the following Dirichlet series of $s$:

$\zeta (s)=\sum _{n=1}^{\infty }{\frac {1}{n^{s}}}$.

In other words, it is the Dirichlet series for the all ones function $U$.

It can also be expressed as a product, using the product formula for Dirichlet series of completely multiplicative function:

$\zeta (s)=\sum _{p\in \mathbb {P} }{\frac {1}{1-p^{-s}}}$.

As the function obtained by analytic continuation

The Dirichlet series for the Riemann zeta-function is absolutely convergent for all complex numbers $s$ for which $\operatorname {Re} (s)>1$. Although the series does not make sense for other $s$, the function extends to a meromorphic function of $\mathbb {C}$, with a single simple pole at the point $1$.

In terms of the Dirichlet eta-function

The Riemann zeta-function can be defined in terms of the Dirichlet eta-function:

$\zeta (s)={\frac {\eta (s)}{1-2^{1-s}}}$.

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.

Related functions

Similarly defined functions

• Dirichlet L-function
• Dedekind zeta-function is a generalization to number fields.

Other related functions

• Gamma function
• Beta function

Zeros and poles

Poles

The Riemann zeta-function has a single simple pole at $1$.

Zeros

The Riemann zeta-function has zeros at all negative even integers. There are also infinitely many zeros with real part $1/2$. The Riemann hypothesis states that all zeros other than the negative even integers have real part $1/2$.