I have a question about convergence and properties of Dirichlet series. it seems a bit interesting and different about the convergences of Dirichlet Series to me.


With $c\in [0,1]$, 


$$f(n) = \pm 1,\qquad |\sum\_{n\le x} f(n)-x^c|\le 2$$


Let A(x)= $\sum\_{n\le x} f(n)$ be a Summatory Function.And F(S)=$\sum\_n f(n)n^{-s}$ be a Dirichlet series.


**So Why and How;**


$\sum\_n f(n) n^{-s}$ converges for $\Re(s) > c$ and $\sum\_n |f(n)|n^{-s}$ converges for $\Re(s) > 1$ ? 


Here is a referrence <https://math.stackexchange.com/questions/1962712/polynomial-growth>. i tried to solve it by using this theorem in this link.I also have looked a lot of books and article , and have searched to find it.


In general , i look at Analytic Number theory and Multiplicative Number Theory, but I couldn't see the form and way of solution of this question. May you help me ? you can also suggest me referrences or hints.Thanks.



Properties of Dirichlet series
Integrating/summing by parts allows you to prove that if $\sum A(n)/n^{s+1}$ converges then $\sum f(n)/n^s$ converges, which gives you the requested convergence. In general, the region where a Dirichlet series is conditionally convergent but not absolutely convergent can be up to width 1, but no larger.


Titchmarsh's Theory of Functions has a thorough chapter on Dirichlet series with many results like these (it also comes up in his Theory of the Riemann Zeta Function)


