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
\(\zeta(s) = \sum_{k=1}^\infty k^{-s}\).
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gsl_sf_zeta_int(n)
This routine computes the Riemann zeta function \(\zeta(n)\) for integer
\(n, n \ne 1\).
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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.
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gsl_sf_zetam1_int(n)
This routine computes \(\zeta(n) - 1\) for integer \(n, n \ne 1\).
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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
\(\zeta(s,q) = \sum_0^\infty (k+q)^{-s}\).
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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 \(\eta(s) = (1-2^{1-s}) \zeta(s)\).
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gsl_sf_eta_int(n)
This routine computes the eta function \(\eta(n)\) for integer \(n\).
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gsl_sf_eta(s)
This routine computes the eta function \(\eta(s)\) for arbitrary \(s\).