Convergence of Dirichlet Series with Bounded Partial Sums

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

Suppose that there exists $B > 0$ such that for all $n, m \in \N$


 * $\displaystyle \left|{\sum_{k \mathop = m}^n a_n}\right| \le B$

Then the Dirichlet series $\displaystyle f(s) = \sum_{n \mathop \ge 1} a_n n^{-s}$ converges locally uniformly to an analytic function on $\Re(s) > 0$.

Proof
By Exponential is Entire, the partial sums


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

are analytic.

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

For $0 < A < \pi / 2$, $\delta > 0$ we let


 * $D_{A,\delta} = \{ s \in \C : -A < \arg(s) < A,\ \Re(s) > \delta$

Then for any $s \in \C$ such that $\Re(s) > 0$, we can choose $A$, $\delta$ such that $s \in D_{A,\delta}$.

So it is sufficient to prove locally uniform convergence in this region.

Also note that if $\lambda_n = \log n$, then


 * $\displaystyle f(s) = \sum_{n \mathop \ge 1} a_n e^{-\lambda_n s}$

By Abel's Lemma, for $N, M \in \N$ we have


 * $\displaystyle \sum_{n \mathop = M}^N a_n e^{-\lambda_n s} = \sum_{n \mathop = M}^{N-1} \left({\sum_{k \mathop = M}^n a_n}\right) \left[{e^{-\lambda_n s} - e^{-\lambda_{n+1} s}}\right] + e^{-\lambda_N s} \left({\sum_{n \mathop = M}^N a_n}\right)$

Let $\epsilon > 0$ be arbitrary, and choose $N_0 \in \N$ such that $\displaystyle \left|{e^{-\lambda_n s}}\right| < \epsilon$ for all $n \ge N_0$.

Now for $N, M \ge N_0$, by the above we obtain


 * $(1): \quad \displaystyle \left\vert \sum_{n \mathop = M}^N a_n e^{-\lambda_n s} \right\vert \le B \sum_{n \mathop = M}^{N-1} \left\vert e^{-\lambda_n s} - e^{-\lambda_{n+1} s} \right\vert + B \epsilon$

For any $\alpha, \beta \in \R$ we have

Therefore,

Finally letting $\Im(s) = t$ we have


 * $\dfrac{|s|} \sigma = \dfrac{\sqrt{\sigma^2 + t^2}} \sigma = \sqrt{1 + \dfrac{t^2}{\sigma^2}}$

and


 * $\dfrac{\pi}2 > A \ge \arg(s) = \arctan \left( \dfrac t \sigma \right)$

so $\dfrac t \sigma < \tan A$ is bounded uniformly in $D_{A,\delta}$.

Thus letting $N \to \infty$, we have shown that:


 * $\displaystyle \left\vert \sum_{n \mathop = M}^\infty a_n e^{-\lambda_n s} \right\vert \to 0$

as $M \to \infty$ uniformly in $D_{A,\delta}$.