zeta

→Tensor

zeta(x: Tensor, q: Tensor)

source

Hurwitz zeta function $\zeta(x, q)$ .

Computes the Hurwitz zeta function, a two-argument generalisation of the Riemann zeta that arises in moment-generating functions of discrete distributions, in expansions of polygamma functions, and as a regulariser in analytic number theory.

Parameters

xTensor

Exponent; must satisfy

\Re(x) > 1

for the series to converge. Any floating-point dtype.

qTensor

Shift parameter; broadcast-compatible with x. Should be positive (avoid the poles at non-positive integers).

Returns

Tensor

$\zeta(x, q)$ element-wise, broadcast to the common shape of x and q.

Notes

Series definition:

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

The implementation accumulates an explicit prefix of 12 terms and closes the tail with the Euler–Maclaurin correction

\sum_{k=N}^\infty (k + q)^{-x} \approx \frac{a^{1-x}}{x - 1} + \frac{a^{-x}}{2} + \sum_j \frac{B_{2j}}{(2j)!} \prod_{i=0}^{2j-2}(x + i)\, a^{-x-2j+1},

where $a = q + N$ and the $B_{2j}$ are Bernoulli numbers. Accuracy is roughly $10^{-6}$ for $x > 1$ and moderate q — adequate for ML log-density work, not for heroic-precision scientific computing. The Riemann zeta is the special case $\zeta(x) = \zeta(x, 1)$ .

Examples

>>> import lucid
>>> from lucid.special import zeta
>>> x = lucid.tensor([2.0, 3.0])
>>> q = lucid.tensor([1.0, 1.0])
>>> zeta(x, q)
Tensor([1.6449, 1.2021])