Zeta functions
--------------

This section includes the Riemann zeta functions
and associated functions pertaining to analytic number theory.

.. contents::

Riemann zeta function (:func:`zeta`, :func:`altzeta`)
.....................................................

.. autofunction:: mpmath.functions.zeta(s)
.. autofunction:: mpmath.functions.altzeta(s)

Zeta function zeros (:func:`zetazero`)
......................................

.. autofunction:: mpmath.functions.zetazero(n, url='http://www.dtc.umn.edu/~odlyzko/zeta_tables/zeros1')

Riemann-Siegel functions (:func:`siegelz`, :func:`siegeltheta`, :func:`grampoint`)
..................................................................................

.. autofunction:: mpmath.functions.siegelz(t)
.. autofunction:: mpmath.functions.siegeltheta(t)
.. autofunction:: mpmath.functions.grampoint(n)

Prime counting function (:func:`primepi`, :func:`primepi2`, :func:`riemannr`)
.............................................................................

.. autofunction:: mpmath.functions.primepi(x)
.. autofunction:: mpmath.functions.primepi2(x)
.. autofunction:: mpmath.functions.riemannr(x)

Bernoulli numbers and polynomials (:func:`bernoulli`, :func:`bernfrac`, :func:`bernpoly`)
.........................................................................................

.. autofunction:: mpmath.functions.bernoulli(n)
.. autofunction:: mpmath.functions.bernfrac(n)
.. autofunction:: mpmath.functions.bernpoly(n,z)

Polylogarithms (:func:`polylog`)
................................

.. autofunction:: mpmath.functions.polylog(s,z)

Stieltjes constants (:func:`stieltjes`)
.......................................

.. autofunction:: mpmath.functions.stieltjes(n,a=1)

Bell numbers and polynomials (:func:`bell`, :func:`polyexp`)
............................................................

.. autofunction:: mpmath.functions.bell(n,x)
.. autofunction:: mpmath.functions.polyexp(s,z)

Other zeta functions (:func:`primezeta`)
........................................

.. autofunction:: mpmath.functions.primezeta(s)
