Proving convergence of this particular Dirichlet series

Question

I am learning number theory from Tom M Apostol and trying some exercises in chapter 12 ( page -273) .

I have done part (a) but unable to do part (b) , and as $\sigma$ >0 , I have no clue which result I can use here.

Can you please tell how should I proceed? I don't have any help as I am self studying.

Thanks!!

Answer 1

For $\Re(s) > \sigma$ the abscissa of convergence $$\sum_{n=1}^\infty a_n n^{-s}=\sum_{m=1}^\infty (\sum_{n=1}^m a_n) (m^{-s}-(m+1)^{-s})=\sum_{m=1}^\infty (\sum_{n=1}^m a_n)\int_m^{m+1} s t^{-s-1}dt$$

For the entire-ness, your problem is using that $(s-1)\zeta(s,r)$ is entire, which is not immediate.

With $h(x) = \sum_{n=1}^\infty a_n e^{-nx}$ you can show it from $$\Gamma(s)\sum_{n=1}^\infty a_n n^{-s}=\int_0^\infty h(x)x^{s-1}dx$$ $$=\sum_{k=0}^K \frac{h^{(k)}(0)/k!}{s+k}+\int_0^\infty (h(x)-1_{x<1}\sum_{k=0}^K x^k h^{(k)}(0) /k!)x^{s-1}dx$$ the latter integral being analytic for $\Re(s) > -K-1$ when $h$ is smooth at $0$.