# Dirichlet series represents an analytic function

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 $\Re(s)>a$?

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Hint: Use summation by parts. – Eric Naslund Dec 10 '11 at 18:03
The only thing to check is to see if $F(s)$ converges in the half-plane $Re(s)>a$, right? – Rob Dec 10 '11 at 18:12
Exactly. 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 $\sigma>\sigma_c$. – Eric Naslund Dec 10 '11 at 18:28