Sunday, 14 November 2010

Convergence of a sequence of continuable Dirichlet series

Let's say f is a Dirichlet series which converges on the half-plane textRes>sigma to a function f(s). Suppose further that f(s) admits an analytic continuation to an entire function, together with the standard sort of functional equation. Let gn be a sequence of Dirichlet series, also convergent on textRes>sigma, which each admit an analytic continuation and functional equation, though their precise FEs may vary. We assume that gn converges to f in the following sense: for every m>0 there exists an N for which the series gn and f match on every term up to the mth, for all n>N. Note this implies that gn(s) converges to f(s) for every textRes>sigma.



Can it be said that gn(s) converges to f(s) for any s outside the domain of convergence?



Perhaps that's too much to hope for, and you can't even expect that gn(s) converges to f(s) even for the point s=sigma. I'd certainly be interested in a counterexample which does this!

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