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Dirichlet Series that fail to be L-functions
Define the prime zeta function $P(s)$ as
$$
P(s)=\sum_pp^{-s}.
$$
By Möbius inversion, one can verify that (configure $\log$ by specifying $\log\zeta(+\infty)=0$)
$$
P(s)=\sum_{n\ge1}{\mu(n)\over n}\l …