Common name of $\sum_n 1/(x^p + n^p)$?

Question

Famously, one has that the cotangent can be written as the Fourier sum $$\pi z \cot \pi z = 1 - 2z^2 \sum_{n=1}^\infty \frac{1}{n^2 -z^2}$$ demonstrating evenly spaced poles alone the $x$-axis. Similarly, the hyperbolic cotangent has poles along the imaginary axis: $$\pi z \coth \pi z = 1 + 2z^2 \sum_{n=1}^\infty \frac{1}{n^2 + z^2}$$ What about poles along rays on arbitrary roots of unity? Consider $$ f_p(z) = \sum_{n=1}^\infty \frac{1}{n^p + z^p}$$ for integer $p$. This function clearly has evenly-spaced poles along rays: so, for $p=3$ the poles are on a tri-axial pattern, along the negative x-axis, and along the two rays at 60 degrees.

The question is: does this function have a common name, or related to some "obvious" function with a common name, or generally studied in some particular setting? It clearly participates in a rich set of identities involving Dirichlet characters, cyclotomic polynomials, Ramanujan-style identities, and so on. But I don't recall spotting it in any books on combinatorics.

(Conversely, how about functions with zeros along rays?)

Answer 1

As pointed out in a comment, one can "easily" show that

$$ \sum_{n=1}^\infty \frac{1}{n^p+z^p} = \lim_{s\to 1} \sum_{k=0}^{p-1} e^{i2\pi k/p} \zeta(s, ze^{i2\pi k/p}) $$

where $\zeta(s,a)$ is the Hurwitz zeta function. I suppose that this answer is enough to send me on my merry way, since both the Hurwitz zeta and the roots of unity can be taken to be "well understood" ingredients.