Brainstorming to search links between $\pi(x)$ and the convexity of $-\int_2^x\frac{dt}{\log(t)}$, related to the Second Hardy–Littlewood conjecture

I was thinking what should be a good way (a reasoning), if there is one, to combine the convexity of the function $$-\operatorname{Li}(x)=-\int_2^x\frac{dt}{\log(t)}$$ on the interval $[2,x]$ with the prime-counting function $\pi(x)$ in the context of the Second Hardy–Littlewood conjecture to state some claims or an interesting open question.

Attempt. My idea is to combine with Jensen's inequality, see this Wikipedia. I say maybe for $\varphi(x)=-\int_2^x\frac{dt}{\log(t)}$ and taking $f(x)=\pi(x)$ with the purpose to write an interesting fact or open question in the context the Second Hardy–Littlewood conjecture. I don't know if such idea was in the literature or well if this idea is feasible.

Question. What are your ideas to state an interesting claim or an interesting open question, if is it feasible, combining the convexity of $-\int_2^x\frac{dt}{\log(t)}$ and propositions/questions or heuristics involving the prime-counting function $\pi(x)$ in the context of Second Hardy–Littlewood conjecture? Many thanks.

If you think that combine with Jensen's inequality is a good way feel free to do it, obviously my reasoning is unfinished. If my question was in the literature please refers those papers and I try find and read those papers.

References:

[1] Arguments for the second Hardy–Littlewood conjecture being false?, from MathOverflow.

I believe that the original conjecture due to Hardy and Littlewood was in next reference:

[2] Hardy and Littlewood, Some Problems of 'Partitio Numerorum.' III. On the Expression of a Number as a Sum of Primes, Acta Math. 44, 1-70, (1923).

• I hope that the spirit of this question is well received. Feel free to add heuristics or calculations with your computer in your answer to my Question. Many thanks.
– user243301
Commented Sep 23, 2017 at 11:58
• Tu fais exprès de ne pas savoir écrire clairement une question ? $\text{Li}(x)$ est convexe et alors ? Commented Sep 23, 2017 at 15:18
• @reuns comment ça va? I am agree that maybe this question is unclear, thus you or other users feel free to criticize it. On the other hand I am interesting if it is possible to combine the convexity of the logarithmic integral and the Second Hardy–Littlewood conjecture. What is the first steps to explore it (if it is feasible)? Many thanks for your comments. And my question is a brainstorming thus you can not ask me for hints :)
– user243301
Commented Sep 23, 2017 at 15:33
• blablabla ${}{}{}{}$ Commented Sep 23, 2017 at 15:42
• No mathematical question, no mathematical answer. Commented Sep 23, 2017 at 15:55

The problem here and in the ref [1] you mentioned is that $Li(x)$ is very smooth increasing function with $\frac{1}{\ln x}$ is decreasing, so to put it in words , the function $Li(x)$ is from the heaven of functions on the other side $\pi(x)$ is not-smooth nor decreasing or increasing and the most difficulty with $\pi(x)$ is that for infinitely many times $\pi(x) >> Li(x)$ and for infinitely many times $\pi(x) << Li(x)$.
Also its hard work and if its going to tell you something, i bet its about $Li(x)$ (which we do know a lot about) and not $\pi(x)$ (which we don't).