# Zeta Functions¶

The Riemann zeta function is defined in Abramowitz & Stegun, Section 23.2.

## Riemann Zeta Function¶

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

gsl_sf_zeta_int(n)

This routine computes the Riemann zeta function $$\zeta(n)$$ for integer $$n, n \ne 1$$.

gsl_sf_zeta(s)

This routine computes 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.

gsl_sf_zetam1_int(n)

This routine computes $$\zeta(n) - 1$$ for integer $$n, n \ne 1$$.

gsl_sf_zetam1(s)

This routine computes $$\zeta(s) - 1$$ for arbitrary $$s, s \ne 1.$$.

## Hurwitz Zeta Function¶

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

gsl_sf_hzeta(s, q)

This routine computes the Hurwitz zeta function $$\zeta(s,q)$$ for $$s > 1, q > 0$$.

## Eta Function¶

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

gsl_sf_eta_int(n)

This routine computes the eta function $$\eta(n)$$ for integer $$n$$.

gsl_sf_eta(s)

This routine computes the eta function $$\eta(s)$$ for arbitrary $$s$$.