Riemann zeta-function

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Definition

As a Dirichlet series

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

.

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

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.

Related functions

Similarly defined functions

Other related functions

Zeros and poles

Poles

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

Zeros

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 .