I know that(I can't find reference for the moment) there are theorems which gives the asymptotic of the summatory functions of the following Dirichlet series

$$ F(s)=\sum_{n=1}^\infty\frac{a_{n}}{n^s} $$ converging absolutely for $Re(s)>1$ and around $s=1$(maybe Re(s)<1 exluded) satisfies $$ F(s) \sim c(s-1)^w $$ for $w$ complex, provided that $F(s)(s-1)^{-w}$ is continuos on the line $s=1$. As far as I know from the first theorem of Wiener-Ikehara positivity of the coefficients is assumed. Is there any way to avoid this?