# Convergence of Dirichlet Series with Bounded Coefficients

## Theorem

Let $\left\langle{a_n}\right\rangle_{n \mathop \in \N}$ be a bounded sequence in $\C$.

Then the Dirichlet series:

- $\displaystyle f \left({s}\right) = \sum_{n \mathop \ge 1} a_n n^{-s}$

converges absolutely and locally uniformly to an analytic function on $\Re \left({s}\right) > 1$.

## Proof

By Exponential is Entire, the partial sums:

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

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 $a_n$:

- $\forall n \in \N: \left\vert{a_n}\right\vert \le B$

Let $D$ be any open subset of $\Re \left({s}\right) > 1$.

So for some $\kappa > 0$:

- $\forall s \in D: \Re \left({s}\right) \ge 1 + \kappa$

Now:

\(\displaystyle \left\vert{f_N \left({s}\right)}\right\vert\) | \(\le\) | \(\displaystyle \sum_{n \mathop = 1}^N \left\vert{a_n}\right\vert \left\vert{n^{-s} }\right\vert\) | $\quad$ | $\quad$ | |||||||||

\(\displaystyle \) | \(\le\) | \(\displaystyle B \sum_{n \mathop = 1}^N \frac 1 {n^{1 + \kappa} }\) | $\quad$ | $\quad$ |

which we know to be finite (!).