I was wondering how exactly the summation by parts works to get equation (2) from prime number theorem ? Thanks.

*[This is an instructive exercise and I recommend that you try your hand at it. You can take a look at the proof of Theorem 36 of https://terrytao.wordpress.com/2014/11/23/254a-notes-1-elementary-multiplicative-number-theory/ for some related arguments. -T.]*

*[Corrected, thanks – T.]*

Somewhat like, to see if a living creature is a human being, it’s a good idea to test its DNA, but perhaps it’s not very useful for the understanding of the distribution of people in the world…

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