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 >= 1are 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 - 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) > 1e9and- 0 <= abs(z.real) < 2.5are 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 - polygammafunction:- >>> m = 3 >>> x = 1.25 >>> polygamma(m, x) array(2.782144009188397) >>> (-1)**(m+1) * factorial(m) * zeta(m+1, x) 2.7821440091883969