Continuation of Dirichlet series I

We study Dirichlet series arising as linear functionals on an inner product space of meromorphic functions and establish a relation between the discontinuities of the former on the boundary and the poles and zeros of the latter on the imaginary axis. As an example application of Delange’s Tauberian theorem, it is shown that the conjectured asymptotic in the additive divisor problem follows conditionally on the non-vanishing of certain meromorphic functions on the imaginary axis.