First let me explain that a big part of the research on this topic was done by using Wolfram. And if this becomes an interesting discovery Wolfram is there - in References. I had some problems with bigger numbers, i.e. calculating Riemann for numbers larger than 10^200. But may be this is a normal limit of the resources used.

Now I’m sharing results from a new formula I developed for approximating the prime counting function π(x), which uses a floating logarithmic base instead of the standard ln(x). The formula achieves better accuracy than Gauss’s approximation, and in some intervals even outperforms Li(x) — while remaining computationally simpler.
The preprint is available on Zenodo:

https://zenodo.org/records/15353321

Numerical tests were conducted up to values 10^12. The formula is within Dusart intervals up to 10^1000. In fact a later test after the preprint post shows values within Dusart even at 10^100 000 000. I welcome feedback, critique, or ideas on a possible theoretical justification behind the observed accuracy.