Convergence of Dirichlet Series with Bounded Coefficients

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Theorem

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

Then the Dirichlet series:

$\ds \map f s = \sum_{n \mathop \ge 1} a_n n^{-s}$

converges absolutely and locally uniformly to an analytic function on $\map \Re s > 1$.


Proof

By Exponential is Entire, the partial sums:

$\ds \map {f_N} s = \sum_{n \mathop = 1}^N a_n n^{-s}$

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 $a_n$:

$\forall n \in \N: \size {a_n} \le B$

Let $D$ be any open subset of $\map \Re s > 1$.

So for some $\kappa > 0$:

$\forall s \in D: \map \Re s \ge 1 + \kappa$


Now:

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

which we know to be finite (!).