Continuation of Dirichlet series I
Number Theory
2025-10-22 v3
Abstract
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.
Cite
@article{arxiv.2510.06523,
title = {Continuation of Dirichlet series I},
author = {Kevin Smith},
journal= {arXiv preprint arXiv:2510.06523},
year = {2025}
}
Comments
In v3 an oversight in v2 is corrected, ensuring that linearity of the main result holds