On the abscissas of a Dirichlet series and its subseries supported on prime factorization
Chaman Kumar Sahu
TL;DR
The paper investigates the relationship between the abscissa of absolute convergence $\sigma_a(f)$ and the restricted-subseries abscissa $\delta_a(f)$ for Dirichlet series with positive coefficients, and proves that the difference $\sigma_a(f)-\delta_a(f)$ can attain any nonnegative value $r$. It provides explicit prime-based constructions of Dirichlet series achieving $\sigma_a(f)=r$ and $\delta_a(f)=0$, including a version with multiplicative coefficients for $r\ge 1$, as well as an alternative approach for $r=0$. The authors then apply these results to the multiplier algebras of Hilbert spaces associated with diagonal Dirichlet-series kernels, obtaining precise containment relations and, in certain cases, equalities with classical multiplier algebras. These findings clarify the range of possible abscissa differences and illuminate the structure of multiplier algebras in Dirichlet-series settings.
Abstract
For a sequence $\{a_n\}_{n \geq 1} \subseteq (0, \infty)$ and a Dirichlet series $f(s) = \sum_{n=1}^\infty a_n n^{-s},$ let $σ_a(f)$ denote the abscissa of absolute convergence of $f$ and let \begin{equation} δ_a(f): = \inf\Bigg\{\Re(s) : \sum\limits_{\substack{j= 1 \\ \tiny{\mbox{gpf}}(j) \leq p_n }}^\infty a_j j^{-s} < \infty ~\text{for all}~ n \geq 1\Bigg\}, \end{equation} where $\{p_j\}_{j \geq 1}$ is an increasing enumeration of prime numbers and $\text{\bf gpf}(n)$ denotes the greatest prime factor of an integer $n \geq 2.$ One significant aspect of these abscissas is their crucial role in analyzing the multiplier algebra of Hilbert spaces associated with diagonal Dirichlet series kernels. The main result of this paper establishes that $σ_a(f)- δ_a(f)$ can be made arbitrarily large, meaning that it can be equal to any non-negative real number. As an application, we determine the multiplier algebra in some cases and, in others, gain insights into the structure of the multiplier algebra of certain Hilbert spaces of Dirichlet series.