Doubt in a proof related to Dirichlet series in Analytic number theory

Question

I am studying Analytic number theory from Tom M Apostol introduction to analytic number theory.

I have doubt in theorem 11.13 whose image I am posting-

I have these 2 doubts.

Doubt 1 How can Apostol assume that F(s) is analytic at a= c +1

I know F is analytic in some disk about s=c, but how to be sure that such a disk has radius of convergence > 1 .

Doubt 2 how to deduce using this theorem the statement given in 5 th line of statement of theorem which is --> if Dirichlet series has a finite abcissa of absolute convergence $\sigma_c$ , then F(s) has a singularity on real axis at point s= $\sigma_c$ .

Can someone please explain.

Answer 1

The function is analytic on $\Re(s) > c$ and on a disk around $c$ thus it is analytic on a disk of radius $1+\epsilon$ around $1+c$ thus (Cauchy integral formula) on this disk it is represented by its Taylor series at $1+c$.

Since $F(s)=\sum_{n\ge 1}f(n)n^{-s},f(n)\ge 0$ all the terms of $$F(c-\epsilon)=\sum_{k\ge 0} (-1-\epsilon)^k \frac{F^{(k)}(1+c)}{k!}$$ are non-negative thus we can change the order of summation to obtain that $$F(c-\epsilon)=\sum_{n\ge 1}f(n)n^{-c+\epsilon}\quad converges$$

Whence Dirichlet series (more generally Mellin/Laplace transforms) with non-negative coefficients have a singularity at $\sigma$ their abscissa of convergence