Explicit formulae for L-function

1. Aug 24, 2008

mhill

let be the L-function $$F(s)= \sum_{n=1}^{\infty} X(n) n^{-s}$$ with a single pole at s=1

then my question is if one can define $$\frac{-F'(s)}{F(s)}= \sum_{n=1}^{\infty} a(n) n^{-s}$$,

then taking an inverse Mellin transform we get

$$\sum_{n \le x} a(n) = x - \sum_{r} r^{-1} x^{r}$$

the question is , what are the a(n) ,

if X(n)=1 for every n then F(s) is just Riemann zeta and a(n) /\(n) Mangoldt function,

the question is , can the result be generalized for every X(n)