Let $T(x)=\sum_{n \leq x} t_n$ and $T(X)=O(x^a)$ for $a \geq 0$. Now let $F(s)=\sum_{n=1}^{\infty} \frac{t_n}{n^s}$ What needs to be checked to prove that this Dirichlet series represents an analytic function in the half plane $\Re(s)>a$?
Dirichlet series represents an analytic function
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analytic-number-theory
dirichlet-series
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3Exactly. You do need uniform convergence to show that it is analytic, and there is a Lemma you can prove which states that if $F(s)$ converges at $s_0=\sigma_0+it_0$ then it converges uniformly in the sector sector $\{s:\ \sigma\geq \sigma_0,\ |t-t_0|\leq H|\sigma-\sigma_0|\}$ for any H>0. Using summation by parts you can show that $\sigma_c =\limsup_{x\rightarrow \infty} \frac{\log |T(x)|}{\log x}$ where $\sigma_c$ is the abscissa of convergence, and the from here reason that $F(s)$ is analytic in the half plane \sigma>\sigma_c. – 2011-12-10