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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?

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  • $\begingroup$ I agree with you that positivity of coefficients is often assumed. If you know something about your dirichlet series past s=1 then the Selberg-Delange method might work. A good reference for all these Tauberian theorems is Tenenbaum's book "An introduction to analytic and probabilistic number theory" $\endgroup$
    – Daniel Loughran
    Dec 27, 2013 at 11:37
  • $\begingroup$ In fact my starting point was Selberg-Delange method. But once I try to factor the complex power of the zeta function, it turns out that one of the factors grows exponentially along to vertical lines inside of the critical strip. Thus complex integration method doesn't work. $\endgroup$
    – Yıldırım A.
    Dec 27, 2013 at 12:37

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