rearrangement of Dirichlet series and continuity at the abscissa of convergence

2010-06-19T00:50:53Z

Let $D(s):=\sum_{n\geq 1}a_n e^{-\lambda_n s}$ be a general Dirichlet series of type $(\lambda_n) _{n\geq 1}$ with finite abscissae of convergence and absolute convergence, respectively $\sigma_c$ and $\sigma_a$. Let also $\sigma_c < \sigma_a$. Pick a fixed $s_0\in\mathbb{C}$ with $\sigma_c <\Re(s_0)<\sigma_a$. Then $\sum_{n\geq 1}a_n e^{-\lambda_n s_0}$ is conditionally convergent. Now, according to a theorem of Steinitz (generalization of Riemann series theorem to finite-dimensional vector spaces over the reals)* there is an affine subspace $A\subset\mathbb{C}$ such that $$ \forall w\in A\ \exists \tau\in S(\mathbb{N}):\ \sum_{n\geq 1}a_{\tau(n)} e^{-\lambda_{\tau(n)} s_0}=w, $$ where $S(\mathbb{N})$ denotes the set of all auto-bijections of the natural numbers ("rearrangements"). We discuss the case when $A$ is a non-trivial affine subspace (consists of more than just one point), so we can assume that $D(s_0)\neq w$ for some $\tau\in S(\mathbb{N})$. Now, let us observe the Dirichlet series defined by $$ D_{\tau}(s):=\sum_{n\geq 1}a_{\tau(n)} e^{-\lambda_{\tau(n)} s}. $$ The Dirichlet series $D_{\tau}$ is convergent in the point $s_0$ and hence locally uniformly convergent in the open half-plane ${\Re(z)> \Re(s_0)}$, thus being analytic there. However, $D$ and $D_{\tau}$ agree on the open half-plane ${\Re(z)>\sigma_a}$ since absolute convergence is invariant with respect to rearrangements, hence they also agree in the open half-plane ${\Re(z)>\Re(s_0)}$ by analycity. Since $D(s_0)\neq w=D_{\tau}(s_0)$, it follows that $D_{\tau}$ cannot be continuous in $s_0$, although it is convergent in it. Provided that my argumentation has so far no flaws, there remains the only possibility that $\Re(s_0)$ is the abscissa of convergence of $D_{\tau}$. Hence my question:

(Q) Given that a Dirichlet-series converges on its abscisse of convergence, are there any known conditions for it to be also continuous there? How about holomorphic? (Or am I missing something here?)

Note that the exact behavior of Dirichlet series on the abscissa of convergence is in general an open problem.

Thanks in advance!

*see for instance the end of http://en.wikipedia.org/wiki/Riemann_series_theorem as well as http://de.wikipedia.org/wiki/Steinitzscher_Umordnungssatz (only in German)

rearrangement of Dirichlet series and continuity at the abscissa of convergence - MathOverflow

rearrangement of Dirichlet series and continuity at the abscissa of convergence