`scipy.special._basic:zeta`

source: build-install/usr/lib/python3.14/site-packages/scipy/special/_basic.py:3703

Signature

def zeta(x, q = None, out = None)

Summary

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 (complex dtypes are allowed but x must have zero imaginary part) with x >= 1 are supported, except 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).

Notes

The two-argument version is the Hurwitz zeta function

ζ (x, q) = k = 0 \sum \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.

Examples

import numpy as np
from scipy.special import zeta, polygamma, factorial

✓

Some specific values:

zeta(2), np.pi**2/6

✗

(np.float64(1.6449340668482264), 1.6449340668482264)

zeta(4), np.pi**4/90

✗

(np.float64(1.0823232337111381), 1.082323233711138)

First nontrivial zero:

zeta(0.5 + 14.134725141734695j)

✗

np.complex128(8.878113823622034e-16+1.3951266857616299e-15j)

Relation to the `polygamma` function:

m = 3
x = 1.25

✓

polygamma(m, x)
(-1)**(m+1) * factorial(m) * zeta(m+1, x)

✗

np.float64(2.782144009188397)

Aliases

scipy.special.zeta