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Motivation: It's known that there is a constant $0<K$ 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 $0<K$ 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 $\lim a_n=0$.

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How far did you get? –  draks ... Apr 5 '12 at 23:06

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