Dirichlet series $\sum a_n n^{-x}$ where $(a_1+\dots+a_n)/n \rightarrow A$

Question

I am working on some Dirichlet series and I am familiar with the proofs of convergence of the functions $\zeta$ and $\eta$.

And I was wondering if $g(x)=\sum a_n n^{-x}$ was normally convergent on $[a,b]$ where $1<a<b$ under the assumption that $(a_1+\dots+a_n)/n$ converges to a given value A.

Thus I would be able to prove that $g$ is infinitely differentiable on $]1,+\infty[$ (It is my final goal and I can clearly see how to do all the other steps).

Answer 1

Partial summation $$\sum_{n\ge 1} a_n n^{-s}=\lim_{N\to \infty}( A(N)N^{-s}+\sum_{n=1}^{N-1}A(n)(n^{-s}-(n+1)^{-s}))$$ For $s > 1$ using that $n^{-s}-(n+1)^{-s}=\int_n^{n+1} s t^{-s-1}dt\le sn^{-s-1}$ it is clear that it converges if $A(N)/N$ is bounded.