It is well known that $$\eta(s)=\sum\limits_{k=1}^{\infty}\frac{(-1)^{k-1}}{k^s} =(1-2^{1-s})\zeta(s)$$ But I have the wider problem of evaluating the following $$f(s)=\sum\limits_{k=1}^{\infty}\frac{\zeta^{k}}{k^s}$$ where $\zeta$ is a root of unity. So for instance, I would like to know if there is even a closed form solution for $$f(s)=\sum\limits_{k=1}^{\infty}\frac{i^{k}}{k^s}$$
I am interested in these results because I am trying to find the Zeta regularized sum of a particular series which is in a similar form to the ones stated above. Here is the problem in case you are curious.
Evaluate
$$\lim\limits_{s\to 0} \sum\limits_{k=1}^{\infty}\frac{\zeta^{k}}{k^s}(\sum\limits_{d|k}\frac{(-1)^{d-1}}{d})$$
Any suggestions would be helpful
It's very easy to give analytic continuations for $f(s) = \displaystyle \sum_{n \geq 1}\dfrac{a(n)}{n^s}$ when $a(n)$ is a periodic function (for example, $i^n$ of $\zeta^n$ for $\zeta$ a root of unity).
Suppose the coefficients are periodic with period $Q$. Then write
$$\begin{align} f(s) &= \sum_{0 \leq c < Q}a(c)\sum_{n \geq 1} \frac{1}{(c + nQ)^s} \\ &= \sum_{0 \leq c < Q}a(c)\sum_{n \geq 1}\frac{1}{\left(\frac cQ + n\right)^sQ^s}\\ &= \sum_{0 \leq c < Q}\frac{a(c)}{Q^s}\zeta\left(s, \frac cQ\right), \end{align}$$
where $\zeta(s, \frac cQ)$ is the Hurwitz zeta function. And so these Dirichlet series are finite sums of Hurwitz zeta functions, which have known analytic continuations to the whole plane (and are sums of Dirichlet L-Series, and vice versa).
I had originally stated that your interest was in terms of digamma functions, but that's because I was misremembering the special values of Hurwitz zeta functions. It happens to be that the constant term of the Hurwitz zeta function at the pole at $s = 1$ is given by a digamma, and when you have periodic coefficients whose sum over the period is $0$ (like with roots of unity), one has that
$$ f(1) = \sum_{n \geq 1} \frac{a(n)}{n} = -\frac{1}{Q} \sum_{1 \leq c \leq Q} a(c) \psi\left(\frac cQ\right),$$
but this isn't quite what you're looking for. However, if you accept Hurwitz zeta functions as being "closed form", then perhaps this is all you need.