Convergence of Dirichlet Series with Bounded Coefficients

Theorem
Let $\{ a_n \}_{n \in \N}$ be a bounded sequence.

Then the Dirichlet series $\displaystyle f(s) = \sum_{n \mathop \ge 1} a_n n^{-s}$ converges absolutely and locally uniformly to an analytic function on $\Re(s) > 1$.

Proof
By Exponential is Entire, the partial sums


 * $\displaystyle f_N(s) = \sum_{n \mathop = 1}^N a_n n^{-s}$

are analytic.

So by Uniform Limit of Analytic Functions is Analytic it is sufficient to show locally uniform convergence.

Let $B$ be a bound for the $a_n$, i.e. $|a_n| \le B$ for all $n$.

Let $D$ be any open subset of $\Re(s) > 1$, so for some $\kappa > 0$, $\Re(s) \ge 1 + \kappa$ for all $s \in D$.

Now

which we know to be finite (!).