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I am learning the Selberg-Delange method in order to read this article:

https://arxiv.org/abs/1712.09019

It says at the beginning of section 6.2 that they use a weakened version of the general result found in Tenenbaum's book. In the book, the theorem is stated for $x\geq 3$ and in the article, they use it for $x\geq 2$. I have read the proof of the theorem in the book, and I am not sure where we need to use this hypothesis on $x$, and I am not sure why it can be changed to $x\geq 2$, as in the article.

Any hints on these two things? Thanks in advance.

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The difference between $x\ge3$ and $x\ge2$ is, in my opinion, completely unimportant. When $2\le x<3$, the sum being considered is just $a_1+a_2$, and the right-hand side of the asymptotic formula is some continuous function of $x$ plus $O(1)$; this is trivially a true statement.

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