I want, for each $\alpha \in [0,1]$ to construct an example of a Dirichlet series for which $\sigma _ 1 = \alpha$ and $\sigma _0 = 1$ where $\sigma _0$ is the abscissa of convergence and $\sigma _1$ is the abscissa of absolute convergence.
I'm thinking the $\sigma _0 = 1$ condition means when we take the absolute value, the dirichlet series must become something like $\sum ^\infty n^{-s}$, ie f(s) needs to be $(-1)^{something...}$, but stuck on how to affect $\sigma _0$.