scipy.special.zetac#

scipy.special.zetac(x, out=None) = <ufunc 'zetac'>#

Riemann zeta function minus 1.

This function is defined as

\[\zeta(x) - 1 = \sum_{k=2}^{\infty} \frac{1}{k^x}\]

where x > 1. For x < 1 the analytic continuation is computed. For more information on the Riemann zeta function, see [dlmf].

Parameters:

xarray_like of float: Values at which to compute zeta(x) - 1 (must be real).
outndarray, optional: Optional output array for the function results

Returns:

scalar or ndarray: Values of zeta(x) - 1.

See also

zeta

Notes

Array API Standard Support

zetac has support for Python Array API Standard compatible backends in addition to NumPy. 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

For the NumPy backend, this function supports all NumPy ufunc keyword arguments. Other backends may support out, but none of the other ufunc kwargs. out is typically supported for CuPy and PyTorch, but not currently in cases where SciPy relies on a generic Array API implementation or, for PyTorch on CPU, falls back to the NumPy backend. out is never supported for JAX because JAX arrays are immutable. zetac does not currently support out for the PyTorch backend.

See Support for the array API standard for more information.

References

[dlmf]

NIST Digital Library of Mathematical Functions https://dlmf.nist.gov/25

Examples

>>> import numpy as np
>>> from scipy.special import zetac, zeta

Some special values:

>>> zetac(2), np.pi**2/6 - 1
(0.64493406684822641, 0.6449340668482264)

>>> zetac(-1), -1.0/12 - 1
(-1.0833333333333333, -1.0833333333333333)

Compare zetac(x) to zeta(x) - 1 for large x:

>>> zetac(60), zeta(60) - 1
(8.673617380119933e-19, 0.0)