Convergence of Dirichlet Series with Bounded Coefficients
Jump to navigation
Jump to search
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:
\(\ds \size {\map {f_N} s}\) | \(\le\) | \(\ds \sum_{n \mathop = 1}^N \size {a_n} \size {n^{-s} }\) | ||||||||||||
\(\ds \) | \(\le\) | \(\ds B \sum_{n \mathop = 1}^N \frac 1 {n^{1 + \kappa} }\) |
which we know to be finite (!).
![]() | This article needs to be linked to other articles. In particular: I would imagine the statement (!) is proved somewhere, but can't think what it'd be called You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by adding these links. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{MissingLinks}} from the code. |