polynomial growth I'm currently trying to understand the link between $L(s,f)$ having an abscissa of convergence < $\infty$ and the grown of f(n). Here is what I have so far: 1. Let F(x) be the summatory function of f(n), i.e. $\sum_{n \leq x} f(n)$. Then F(x) has polynomial growth if there exists a polynomial P(x) such that |f(x)| $\leq$ p(x) $\forall x \geq1$. 2. The abscissa of convergence is associated with the growth rate of its summatory function. Can I assume that if F(x) has a polynomial growth, then its associated dirichlet series has a finite abscissa of convergence? If so, why?

Theorem 1.3 of Montgomery and Vaughan's **Multiplicative Number Theory, I** states: Let $\sigma_c$ be the abscissa of convergence of $\sum_{n=1}^\infty f(n)n^{-s}$. If $\sigma_c<0$, then $F(x)$ is bounded. If $\sigma_c\ge0$, then $F(x)$ grows roughly like $x^{\sigma_c}$, in that $$ \limsup_{x\to\infty} \frac{\log |F(x)|}{\log x} = \sigma_c. $$ In particular, yes, polynomial growth of $F(x)$ is equivalent to $\sigma_c\
e+\infty$.