Continuation of Dirichlet series I
Kevin Smith
TL;DR
The paper develops a novel framework of inner product spaces $V_k$ and associated linear functionals to study Dirichlet-series representations, linking boundary discontinuities on the critical axis to poles and zeros of the underlying meromorphic data. A symplectic (skew-symmetric) approach shows that non-continuous boundary behavior forces explicit obstructions in a decomposition $F=D+E$, with the obstruction localized to poles or zeros of $E^{(d)}$. This yields a conditional route to the additive divisor problem:, under Delange's Tauberian theorem, the Dirichlet series $F_{h,k}(s)$ either produces the expected asymptotics or demonstrates obstructions that would invalidate the Tauberian conclusion; conditional on the Lindelöf hypothesis for $k>5$, the expected asymptotic $\sum_{n\le x} d_k(n)d_k(n+h) \sim c_{h,k} x (\log x)^{k-1} (\log(x+h))^{k-1}$ holds for all $k$, with the constant $c_{h,k}$ given by an explicit Euler product. The work illuminates how zeros on the imaginary axis influence mean-value results in divisor problems and outlines a path for future connections with zeros on the imaginary axis.
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.