Zeta functions, L-series and polylogarithms

-------------------------------------------



This section includes the Riemann zeta functions

and associated functions pertaining to analytic number theory.



Riemann zeta function

.....................



:func:`zeta`

^^^^^^^^^^^^^^^^^

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





Hurwitz zeta function

.....................



:func:`hurwitz`

^^^^^^^^^^^^^^^^^

.. autofunction:: mpmath.functions.hurwitz(s,a,derivative=0)





Zeta-type L-functions

.....................



:func:`altzeta`

^^^^^^^^^^^^^^^

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



:func:`dirichletl`

^^^^^^^^^^^^^^^^^^

.. autofunction:: mpmath.functions.dirichlet(s,chi,derivative=0)



:func:`primezeta`

^^^^^^^^^^^^^^^^^

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



Zeta function zeros

......................................



:func:`zetazero`

^^^^^^^^^^^^^^^^

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



Riemann-Siegel functions and Gram points

........................................



These functions are used for the study of the Riemann zeta function

in the critical strip.



:func:`siegelz`

^^^^^^^^^^^^^^^^^

.. autofunction:: mpmath.functions.siegelz(t)



:func:`siegeltheta`

^^^^^^^^^^^^^^^^^^^

.. autofunction:: mpmath.functions.siegeltheta(t)



:func:`grampoint`

^^^^^^^^^^^^^^^^^

.. autofunction:: mpmath.functions.grampoint(n)



Stieltjes constants

...................



:func:`stieltjes`

^^^^^^^^^^^^^^^^^

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



Polylogarithms

................................



:func:`polylog`

^^^^^^^^^^^^^^^

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



Clausen functions

.................



:func:`clsin`

^^^^^^^^^^^^^^^^^

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



:func:`clcos`

^^^^^^^^^^^^^^^^^

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



Polyexponentials

..................................



:func:`polyexp`

^^^^^^^^^^^^^^^^^

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