Given a sequence $a\_n$ such that $\sum\_{n\ge1} \dfrac{|a\_n|}{n^s}$ is convergent for $s>0$. Given that $\sum\_{n\ge1} \frac{|a\_n|}{n} < 1$, is it be possible to impose some sort of an upperbound for $\sum\_{n\ge1} \frac{|a\_n|}{n^s}$ for $1>s>k>0$ for some fixed $k$?



Bounds on Riemann Zeta type Function?
Given an integer $m$, define a sequence $a\_n$ to be $m$ at the $m$-th place and $0$ otherwise.
Then
$$\sum\_{n\ge 1} {a\_n\over n^s}=m^{1-s}$$
As any upper bound must work for all $m$, we see that no upper bound is possible for $s<1$.


