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I'm trying to figure out a Dirichlet series which has its abscissa of absolute convergence $=\frac{1}{2}$. I've been trying to think about using the formula for this abscissa:

$$\sigma_a=\limsup_{n\to\infty}\frac{\log(|a_1|+|a_2|+\cdots+|a_n|)}{\lambda_n}.$$ if $\sum |a_k|$ diverges.

$$\sigma_a=\limsup_{n\to\infty}\frac{\log(|a_{n+1}|+|a_{n+2}|+\cdots)}{\lambda_n}$$ if $\sum |a_k|$ converges.

Thank you in advance.

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  • $\begingroup$ it is the same as the radius of convergence for power series. note that with $s = -\ln z$, a Dirichlet series becomes $\sum_n a_n z^{\ln n}$ and applying the same idea as the radius of convergence, you get the formula above $\endgroup$ – reuns May 24 '16 at 15:44

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