@cindex Zeta functions

The Riemann zeta function is defined in Abramowitz & Stegun, Section
23.2.  The functions described in this section are declared in the
header file @file{gsl_sf_zeta.h}.

@menu
* Riemann Zeta Function::
* Riemann Zeta Function Minus One::
* Hurwitz Zeta Function::
* Eta Function::
@end menu

@node Riemann Zeta Function
@subsection Riemann Zeta Function

The Riemann zeta function is defined by the infinite sum 
@c{$\zeta(s) = \sum_{k=1}^\infty k^{-s}$}
@math{\zeta(s) = \sum_@{k=1@}^\infty k^@{-s@}}.  

@deftypefun double gsl_sf_zeta_int (int @var{n})
@deftypefunx int gsl_sf_zeta_int_e (int @var{n}, gsl_sf_result * @var{result})
These routines compute the Riemann zeta function @math{\zeta(n)} 
for integer @var{n},
@math{n \ne 1}.
@comment Domain: n integer, n != 1
@comment Exceptional Return Values: GSL_EDOM, GSL_EOVRFLW
@end deftypefun

@deftypefun double gsl_sf_zeta (double @var{s})
@deftypefunx int gsl_sf_zeta_e (double @var{s}, gsl_sf_result * @var{result})
These routines compute the Riemann zeta function @math{\zeta(s)}
for arbitrary @var{s},
@math{s \ne 1}.
@comment Domain: s != 1.0
@comment Exceptional Return Values: GSL_EDOM, GSL_EOVRFLW
@end deftypefun


@node Riemann Zeta Function Minus One
@subsection Riemann Zeta Function Minus One

For large positive argument, the Riemann zeta function approaches one.
In this region the fractional part is interesting, and therefore we
need a function to evaluate it explicitly.

@deftypefun double gsl_sf_zetam1_int (int @var{n})
@deftypefunx int gsl_sf_zetam1_int_e (int @var{n}, gsl_sf_result * @var{result})
These routines compute @math{\zeta(n) - 1} for integer @var{n},
@math{n \ne 1}.
@comment Domain: n integer, n != 1
@comment Exceptional Return Values: GSL_EDOM, GSL_EOVRFLW
@end deftypefun

@deftypefun double gsl_sf_zetam1 (double @var{s})
@deftypefunx int gsl_sf_zetam1_e (double @var{s}, gsl_sf_result * @var{result})
These routines compute @math{\zeta(s) - 1} for arbitrary @var{s},
@math{s \ne 1}.
@comment Domain: s != 1.0
@comment Exceptional Return Values: GSL_EDOM, GSL_EOVRFLW
@end deftypefun


@node Hurwitz Zeta Function
@subsection Hurwitz Zeta Function

The Hurwitz zeta function is defined by
@c{$\zeta(s,q) = \sum_0^\infty (k+q)^{-s}$}
@math{\zeta(s,q) = \sum_0^\infty (k+q)^@{-s@}}.

@deftypefun double gsl_sf_hzeta (double @var{s}, double @var{q})
@deftypefunx int gsl_sf_hzeta_e (double @var{s}, double @var{q}, gsl_sf_result * @var{result})
These routines compute the Hurwitz zeta function @math{\zeta(s,q)} for
@math{s > 1}, @math{q > 0}.
@comment Domain: s > 1.0, q > 0.0
@comment Exceptional Return Values: GSL_EDOM, GSL_EUNDRFLW, GSL_EOVRFLW
@end deftypefun


@node Eta Function
@subsection Eta Function

The eta function is defined by
@c{$\eta(s) = (1-2^{1-s}) \zeta(s)$}
@math{\eta(s) = (1-2^@{1-s@}) \zeta(s)}.

@deftypefun double gsl_sf_eta_int (int @var{n})
@deftypefunx int gsl_sf_eta_int_e (int @var{n}, gsl_sf_result * @var{result})
These routines compute the eta function @math{\eta(n)} for integer @var{n}.
@comment Exceptional Return Values: GSL_EUNDRFLW, GSL_EOVRFLW
@end deftypefun

@deftypefun double gsl_sf_eta (double @var{s})
@deftypefunx int gsl_sf_eta_e (double @var{s}, gsl_sf_result * @var{result})
These routines compute the eta function @math{\eta(s)} for arbitrary @var{s}.
@comment Exceptional Return Values: GSL_EUNDRFLW, GSL_EOVRFLW
@end deftypefun