Motivation: It's known that there is a constant $0 such that for any natural number $N$, $KN\leq \frac{\varphi(1)}{1}+\frac{\varphi(2)}{2}+\cdots+\frac{\varphi(N)}{N}$ (with $\varphi$ being Euler's function). See more details here: On sums involving Euler's totient function
My intention is to generalize this result.
So my question is: Suppose that $\{a_n\}_{n=1}^{\infty}$ is a non-increasing sequence of positive reals, is there a constant $0 such that $K(a_1+\cdots+a_N)\leq {\frac{\varphi(1)}{1}}a_1+{\frac{\varphi(2)}{2}}a_2+\cdots+{\frac{\varphi(N)}{N}}a_N$ for every natural number $N$?
Remark: if $\lim a_n>0$, then we can simply take $K=\frac{K'\lim a_n}{a_1}$ where $K'$ is the constant appearing in the first result stated above, so the problem is really when $\lim a_n=0$.
elementary-number-theory
modular-arithmetic
totient-function
arithmetic-functions