Continuous functional calculus on U enabling analytic continuation of Dirichlet series beyond Re(s)>1

Determine whether the Banach space U of arithmetic functions, equipped with the norm ||f||_U = sup_{n in N} |f(n)| / log(2+n), admits a continuous functional calculus that allows analytic continuation of the Dirichlet series D(f;s) beyond Re(s) > 1 for a dense subalgebra of U.

Background

Within the paper, Dirichlet series associated with elements of U are shown to converge absolutely for Re(s)>1, and certain growth conditions yield further holomorphic extension. A functional calculus on U could systematize and extend analytic continuation properties beyond Re(s)>1 in a controlled, operator-theoretic way.

The authors explicitly pose whether such a continuous functional calculus exists for U and can be used to continue D(f;s) past the classical convergence boundary for a dense subalgebra, indicating a structural open problem at the interface of operator theory and Dirichlet series.