OffSet Logarithmic Integral

Hello, I've posted a thread several threads ago about some "intro" questions to a problem and I'm now trying to deal with the last questions of the problem itself.

We are studying

What I've managed (some of it with your help) to prove :

I've also proven that f is a bijection of ]1,+∞[ on ]-∞,+∞[

Here are the questions that are a problem to me : find an equivalent of

I'm not sure if I can directly say that it is directly because of the second relation I've written.

Then I've pretty much have some questions about the bijection which are to study the convergence of those series with different alphas.

Which is, I believe a series of Bertrand (which I've already studied so it's all good) and

If you've got any sort of help with the equivalent and the series of the bijection (which I am totally unable to find because I believe that we cannot express the bijection of f with "classic" functions.

Thank you very much and sorry for my bad Latex, I'm a beginner ^^