On the abscissas of a Dirichlet series and its subseries supported on prime factorization

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^{-\Re(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.