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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.

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Are you asking about Dirichlet series, or twisted Dirichlet series? My guess is both should use the same tools so you should start with usual Dirichlet series (which are covered in many analytic number theory books). If your coefficients are positive, the alternating signs might make the series converge on a bigger half plane (that's the only difference I can see so far). A good place to start might be to study the case $a_n = 1$ (which gives you the $\zeta$ function, and $L$ series in the twisted case). – Joel Cohen May 15 '11 at 1:17

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.

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