Improvement of Gauss's Formula for the distribution of primes by introducing a floating logarithmic base and empirically proven accuracy similar to Li

Description

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 elementary operations, avoiding the complexity of Riemann's approach. Numerical tests up to 10^12 demonstrate superior performance compared to Gauss's classical formula and near-Riemann accuracy. Additional forecasts are provided for 10^100, 10^200, and tests against Dusart intervals at 10^500 and 10^1000. This method opens new possibilities for practical applications and theoretical exploration in analytic number theory.

Categories

Licence