Cubist
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- Oct 29, 2019
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Posted on behalf of @dobribozhilov...
A New Approximation for π(x) — More Accurate than Gauss, Lighter than Li(x)
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:
zenodo.org
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.
A New Approximation for π(x) — More Accurate than Gauss, Lighter than Li(x)
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:
Improvement of Gauss's Formula for the distribution of primes by introducing a floating logarithmic base and empirically proven accuracy similar to Li
This paper proposes an empirical improvement of Gauss's formula for estimating the number of prime numbers, introducing a floating logarithmic base that significantly increases accuracy. The new formula achieves precision close to that of the logarithmic integral Li(x), while using only...
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.