Why is it true that $$\frac{\varphi(m)}{m}\sum_{n \leq x}\frac{1}{n} \leq \sum_{n \leq x, (n, m) = 1}\frac{1}{n}?$$ Intuitively to see this, one can think of that from 1 up to $m$, there are $\varphi(m)$ integers which are relative prime to $m$, so one can expect this to be the proportion, but how does one show this explicitly?
Summation over relatively prime numbers
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number-theory
summation
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1Why do you believe this is true? Where does it come from? – 2012-11-26
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0Put it this way: can you prove it if $m$ is a prime or a prime power? Plus, where did you find it? – 2012-11-27