Proof Janusz Algebraic number fields, convergence of Dirichlet Series.

Question

The book Algebraic number fields, Janusz

Please, Could you explain the proof of the part b) a little more?

Thank you all.

Answer 1

Note that

$$\text{Re}\,s:=\sigma>b\implies \left|\sum_{n=1}^\infty\frac{a(n)}{n^s}\right|\le\sum_{n=1}^\infty\frac{|a(n)|}{n^{\sigma}}\le\sum_{n=1}^\infty\frac{|a(n)|}{n^b}$$

and now use (a) and the fact the last series is uniformly convergent and etc.

I suppose Janusz is assuming Weierstrass M-test or something like that to deduce that a uniformly convergent series of analytic functions is analytic, or may you can prove it directly as it is not, if I recall correctly, that hard.