Sequences and Series/Dirichlet‒Hurwitz series

Definition (Dirichlet‒Hurwitz series):

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

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

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

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

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

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

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

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