Suppose $x>0$ and let $f(x)=\sum_{k\le x}\frac{1}{\varphi(k)}$, where $\varphi(k)$ is the Euler totient function. It is well known that $\sum_{k\le x}\frac{1}{k}\sim\log x$. What is the asymptotic behavior of the sum $f(x)=\sum_{k\le x}\frac{1}{\varphi(k)}$?
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2$\begingroup$ See Wikipedia $\endgroup$– WojowuCommented May 1, 2022 at 14:12
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Here is a question that addresses the mentioned asymptotics - https://math.stackexchange.com/questions/2683190/showing-sum-n-leq-x-frac1-phi-n-c-log-x-o1?noredirect=1&lq=1 Roughly, you can proceed by the standard convolution method via approximating $1/\varphi(n)$ as $1/n$.
