1. ## 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 $\displaystyle$\forall x\geq 1\; f(x)=\int_2^x \dfrac 1{lnt}\; dt$$What I've managed (some of it with your help) to prove : \displaystyle \int_2^x \dfrac 1{({lnt})^2}\; dt=o(f(x))$$

$\displaystyle$f(x)=\dfrac x{lnx}-\dfrac 2{ln2}+\displaystyle\int_{2}^{x} \dfrac 1{({lnt})^2} \, \mathrm dt$$\displaystyle f(x)\sim_{x\to+\infty}\dfrac x{lnx}$$

$\displaystyle$f(x)\sim_{x\to 1} ln(x-1)$$I've also proven that f is a bijection of ]1,+∞[ on ]-∞,+∞[ Here are the questions that are a problem to me : find an equivalent of \displaystyle f(x)-\dfrac x{lnx} when x\rightarrow +\infty I'm not sure if I can directly say that it is \displaystyle -\dfrac 2{ln2}$$ 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.

$\displaystyle$\Displaystyle \sum_{n\geq 3} \dfrac 1{({f(n)})^\alpha} $$Which is, I believe a series of Bertrand (which I've already studied so it's all good) and \displaystyle \Displaystyle \sum_{n\geq 0} \dfrac 1{({{f}^{-1}(n)})^\alpha}$$

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 ^^

2. $\displaystyle$f(x)=\dfrac x{lnx}-\dfrac 2{ln2}+\displaystyle\int_{2}^{x} \dfrac t{({lnt})^2} \, \mathrm dt

$\displaystyle \displaystyle\int_{2}^{x} \dfrac t{({lnt})^2}\mathrm dt=\frac {1}{2}\frac {t^2}{ln^2t} |_2^x+\displaystyle\int_{2}^{x} \dfrac {t^2}{({lnt})^3}\mathrm dt$

This is not useful because free term goes to infinity.
May be this will better
$\displaystyle \displaystyle\int_{2}^{x} \frac {\mathrm dt}{lnt}=t \; ln(lnt)|_2^x-\displaystyle\int_{2}^{x} ln(lnt)\mathrm dt$

3. Hello and thanks for your answer. I don't know how this can help me though since I'm trying to find the equivalent of $\displaystyle$f(x)-\dfrac x{lnx}. Does anyone have an idea on how to find it or can help me with the series?

4. You have:

$\displaystyle \displaystyle f(x)=\int_2^x \dfrac{1}{\ln(t)}dt$

$\displaystyle \displaystyle f(x)=\dfrac{x}{\ln(x)}-\dfrac{2}{\ln(2)}+\int_2^x \dfrac{1}{(\ln(t))^2}dt$

and:

$\displaystyle \displaystyle \dfrac{1}{(\ln(t))^2}dt=o(f(x))$

So:

$\displaystyle \displaystyle f(x)-\dfrac{x}{\ln(x)}=-\dfrac{2}{\ln(2)}+\int_2^x \dfrac{1}{(\ln(t))^2}dt=-\dfrac{2}{\ln(2)}+o(f(x))$

if that is any help?

CB