Convergence of Dirichlet Series with Bounded Coefficients

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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:

\(\displaystyle \left\vert{f_N \left({s}\right)}\right\vert\) \(\le\) \(\displaystyle \sum_{n \mathop = 1}^N \left\vert{a_n}\right\vert \left\vert{n^{-s} }\right\vert\)
\(\displaystyle \) \(\le\) \(\displaystyle B \sum_{n \mathop = 1}^N \frac 1 {n^{1 + \kappa} }\)

which we know to be finite (!).