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

Riemann Zeta Function
---------------------

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

 -- Function: double gsl_sf_zeta_int (int N)
 -- Function: int gsl_sf_zeta_int_e (int N, gsl_sf_result * RESULT)
     These routines compute the Riemann zeta function \zeta(n) for integer N, n \ne 1.

 -- Function: double gsl_sf_zeta (double S)
 -- Function: int gsl_sf_zeta_e (double S, gsl_sf_result * RESULT)
     These routines compute the Riemann zeta function \zeta(s) for arbitrary S, s \ne 1.

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.

 -- Function: double gsl_sf_zetam1_int (int N)
 -- Function: int gsl_sf_zetam1_int_e (int N, gsl_sf_result * RESULT)
     These routines compute \zeta(n) - 1 for integer N, n \ne 1.

 -- Function: double gsl_sf_zetam1 (double S)
 -- Function: int gsl_sf_zetam1_e (double S, gsl_sf_result * RESULT)
     These routines compute \zeta(s) - 1 for arbitrary S, s \ne 1.

Hurwitz Zeta Function
---------------------

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

 -- Function: double gsl_sf_hzeta (double S, double Q)
 -- Function: int gsl_sf_hzeta_e (double S, double Q, gsl_sf_result * RESULT)
     These routines compute the Hurwitz zeta function \zeta(s,q) for s > 1, q > 0.

Eta Function
------------

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

 -- Function: double gsl_sf_eta_int (int N)
 -- Function: int gsl_sf_eta_int_e (int N, gsl_sf_result * RESULT)
     These routines compute the eta function \eta(n) for integer N.

 -- Function: double gsl_sf_eta (double S)
 -- Function: int gsl_sf_eta_e (double S, gsl_sf_result * RESULT)
     These routines compute the eta function \eta(s) for arbitrary S.