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Riemann or Hurwitz zeta function.
Parameters
----------
x : array_like of float or complex.
Input data
q : array_like of float, optional
Input data, must be real. Defaults to Riemann zeta. When `q` is
``None``, complex inputs `x` are supported. If `q` is not ``None``,
then currently only real inputs `x` with ``x >= 1`` are supported,
even when ``q = 1.0`` (corresponding to the Riemann zeta function).
out : ndarray, optional
Output array for the computed values.
Returns
-------
out : array_like
Values of zeta(x).
See Also
--------
zetac
Notes
-----
The two-argument version is the Hurwitz zeta function
.. math::
\zeta(x, q) = \sum_{k=0}^{\infty} \frac{1}{(k + q)^x};
see [dlmf]_ for details. The Riemann zeta function corresponds to
the case when ``q = 1``.
For complex inputs with ``q = None``, points with
``abs(z.imag) > 1e9`` and ``0 <= abs(z.real) < 2.5`` are currently not
supported due to slow convergence causing excessive runtime.
References
----------
.. [dlmf] NIST, Digital Library of Mathematical Functions,
https://dlmf.nist.gov/25.11#i
Examples
--------
>>> import numpy as np
>>> from scipy.special import zeta, polygamma, factorial
Some specific values:
>>> zeta(2), np.pi**2/6
(1.6449340668482266, 1.6449340668482264)
>>> zeta(4), np.pi**4/90
(1.0823232337111381, 1.082323233711138)
First nontrivial zero:
>>> zeta(0.5 + 14.134725141734695j)
0 + 0j
Relation to the `polygamma` function:
>>> m = 3
>>> x = 1.25
>>> polygamma(m, x)
array(2.782144009188397)
>>> (-1)**(m+1) * factorial(m) * zeta(m+1, x)
2.7821440091883969
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