Is there a closed formula for the sum $$\sum \limits_{n=1}^{\infty} \frac{e^{2 \pi i n x}}{n^{a+1}} $$ ? $(a \in (0,1))$
My goal is to show that the series $\sum \limits_{k=1}^{\infty} a_k$ defined by $a_k = \int_0^1 r(x)^k dx$, where $r(x)=\frac{1}{\zeta(1+a)} \sum \limits_{n=1}^{\infty} \frac{e^{2 \pi i n x}}{n^{a+1}}$converges.