Proposition: Let $\left\{ a_{n}\right\} _{n\geq1}$ be a sequence of complex numbers, let: $$A\left(N\right)=\sum_{n=1}^{N}a_{n}$$ and suppose that there is a real number $c>0$ so that $\left|A\left(N\right)\right|=O\left(N^{c}\ln N\right)$ as $N\rightarrow\infty$. Then: $$\sup_{t\in\mathbb{R}}\left|\sum_{n=1}^{\infty}\frac{a_{n}}{n^{\sigma+it}}\right|<\infty,\textrm{ }\forall\sigma>c$$
I can nearly prove this on my own; the only gap is showing that the function: $$F_{\sigma,c}\left(t\right)=\sum_{k=1}^{\infty}\binom{-\sigma-it}{k}\zeta^{\prime}\left(\sigma-c+k\right)$$ satisfies: $$\sigma>c\Rightarrow\sup_{t\in\mathbb{R}}\left|F_{\sigma,c}\left(t\right)\right|<\infty$$
I was wondering if anyone knows of a reference for a proof of this Proposition of mine, or of a way of showing the boundedness condition on $F_{\sigma,c}\left(s\right)$.
