construct an asymptotic, analytic function, which outperforms $\frac{x}{\log(x)}$ but isn't quite as good as $\text{Li}(x)$?

Question

The offset logarithmic integral is defined as $$ \text{Li}(x)=\int_2^x\frac{1}{\log(t)}~dt. $$

It can be shown that $\text{Li}(x)\sim\pi(x)$ where $\pi(x)$ is the prime counting function. It can also be shown that $\frac{x}{\log(x)}\sim \pi(x).$

Is there any practical use in number theory for constructing an asymptotic, analytic function, which outperforms $\frac{x}{\log(x)}$ but isn't quite as good as $\text{Li}(x)$?

Note: $\text {Li}(x)$ is the best possible asymptotic formula to approximate $\pi(x).$

Answer 1

Speaking to the comment by user @Arthur, here is a function that supports their claim that there are "several of these candidate functions" i.e. (better than $\frac{x}{\log(x)})$ but worse than $\text{Li(x)}).$

$$ B_\kappa(x)= \int_2^x \sin\bigg(\frac{\kappa}{\log(t)}\bigg)+\sinh\bigg(\frac{\kappa}{\log(t)}\bigg)~dt.$$ for some $\kappa=0.499998$ and $x=10^6$ one gets $B(10^6)=78,626.2$ compared with $\text{Li(10^6)}=78,627.5$

And $\pi(10^6)=78,498.$

Another example without using an integral is $f(x)=\bigg(\frac{x}{\log(x)}\bigg)\exp\bigg(\frac{1}{\log(x)}\bigg).$