scipy.special.

zeta#

scipy.special.zeta(x, q=None, out=None)[source]#

Riemann or Hurwitz zeta function.

Parameters:

xarray_like of float or complex.: Input data
qarray_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).
outndarray, optional: Output array for the computed values.

Returns:

outarray_like: Values of zeta(x).

See also

zetac

Notes

The two-argument version is the Hurwitz zeta function

\[\zeta(x, q) = \sum_{k=0}^{\infty} \frac{1}{(k + q)^x};\]

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.

Array API Standard Support

zeta has experimental support for Python Array API Standard compatible backends in addition to NumPy. Please consider testing these features by setting an environment variable SCIPY_ARRAY_API=1 and providing CuPy, PyTorch, JAX, or Dask arrays as array arguments. The following combinations of backend and device (or other capability) are supported.

Library	CPU	GPU
NumPy	✅	n/a
CuPy	n/a	✅
PyTorch	✅	✅
JAX	✅	✅
Dask	✅	n/a

See Support for the array API standard for more information.

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