TITLE: Literature Request: Twisted Dirichlet Series
QUESTION [6 upvotes]: Recently, I have been pushed toward studying analytic continuation dirichlet series with twists that are additive.  These are functions $D(s) = \sum e_k(hn)\frac{a_n}{n^s}$ where $a_n$ is some sequence (usually positive) and $e_k(n)=e^{2\pi i \frac{n}{k}}$ with $h\in \mathbb{N}.$
I would like to know if there is any literature on this topic available.

REPLY [3 votes]: I know it is tabbo to answer your own question but the Conclusion is that additive and multiplicative twists same in the sense that if $D(s)$ continues for every $\chi(n)$ which is multiplicative then the additive twist will continue as well.
If $$\tau(\tilde{\chi})=\sum_{n\mod k}\tilde{\chi}(n)e_k(n)$$
is a Gauss sum then $$e_k(n) = \frac{1}{\phi(k)}\sum_{\chi \mod k} \chi(\bar{n}) \tau(\chi) $$ where $\bar{n}n=1\mod k.$  Hence an additive twist can be written as a sum of multiplicative twisted Dirichlet series.