Let $f(s) = \sum_{n=1}^\infty \frac{a_n}{n^s}$ be a Dirichlet series with abscissa of absolute convergence $L$. Then,

1) Is there a formula for $L$ in terms of the coefficients $a_n$?

2) Must $f(s)$ have a singularity somewhere on the line $\Re{s}=L$?


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If $\sum_{n=1}^\infty |a_n| = C$ (if it diverges set $C=0$) then $$L = \lim \sup_{N \to \infty} \frac{\log \left|-C+\sum_{n =1}^N |a_n| \right|}{\log N}$$

$\sum_{n=1}^\infty |a_n| n^{-s}$ converges and is analytic for $\Re(s) > L$ and its analytic continuation (if its exists) has a singularity at $s=L$.

For $\sum_{n=1}^\infty a_n n^{-s}$ all we can say in general is it converges and is analytic for $\Re(s) > L$. See $\sum_{n=1}^\infty (-1)^{n+1} n^{-s}$ which is entire even if $L = 1$.


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