Sequences and Series/Dirichlet‒Hurwitz series

Definition (Dirichlet‒Hurwitz series):

Let ${\displaystyle f:\mathbb {N} \to \mathbb {C} }$ be a function, and let ${\displaystyle a\in \mathbb {C} \setminus \{-1,-2,\ldots \}}$. The Dirichlet‒Hurwitz series associated to ${\displaystyle f}$ and ${\displaystyle a}$ is the function of ${\displaystyle s\in \mathbb {C} }$ given by the series

${\displaystyle \sum _{n=1}^{\infty }{\frac {f(n)}{(n+a)^{s}}}}$.

Definition (abscissa of absolute convergence of Dirichlet‒Hurwitz series):

Let ${\displaystyle f:\mathbb {N} \to \mathbb {C} }$ be a function, and let ${\displaystyle a\in \mathbb {C} \setminus \{-1,-2,\ldots \}}$. Suppose that there exists a number ${\displaystyle \sigma _{a}\in \mathbb {R} }$ such that

${\displaystyle \sum _{n=1}^{\infty }\left|{\frac {f(n)}{(n+a)^{s}}}\right|}$

converges whenever ${\displaystyle \Re s>\sigma _{a}}$ and diverges whenever ${\displaystyle \Re s<\sigma _{a}}$. Then ${\displaystyle \sigma _{a}}$ is called the abscissa of absolute convergence of the Dirichlet‒Hurwitz series associated to ${\displaystyle f}$ and ${\displaystyle a}$.

Proposition (existence of abscissa of absolute convergence of Dirichlet‒Hurwitz series):

Let ${\displaystyle f:\mathbb {N} \to \mathbb {C} }$ be a function, and let ${\displaystyle a\in \mathbb {C} \setminus \{-1,-2,\ldots \}}$. Suppose that