## rearrangement of Dirichlet series and continuity at the abscissa of convergence

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.

 As pointed out by fedja on the AoPS forums, the flaw of the above argumentation is in fact that rearranging the general Dirichlet series breaks the condition $\lambda_n \uparrow \infty$, hence all the standard statements on general Dirichlet series are not anymore applicable in this form. – ex falso quodlibet Jun 29 2010 at 23:38