The Number-Theoretic Foundation of the Modular Sieve Method: An Empirical Study of Dirichlet's Theorem and the Logarithmic

This paper systematically elaborates the number-theoretic founda- tion of the modular sieve method and reveals its intrinsic connection with Dirichlet’s theorem and the logarithmic decay law. The modular sieve method is based on an elementary fact: all primes greater than 5 must lie in the 8 residue classes coprime to 30 (referred to as the ”8 orbits”). By statistically analyzing residue frequencies and apply- ing a dynamic threshold, this method efficiently generates high-purity prime candidate sets and verifies the uniformity of prime distribution among the 8 orbits within the range N = 10⁸. Dirichlet’s theorem (and its precise form—the prime number theorem for arithmetic pro- gressions) not only guarantees infinitely many primes in each orbit but also reveals the logarithmic scale of prime density decaying as 1/ ln x with increasing numerical value. Interestingly, the sieving pro- cess of the modular sieve method also exhibits a profound connection with the natural logarithm: the size of the candidate set S (n) strictly follows S (n) = A−Blnn as the sieving step n increases, with a good- ness of fit as high as R2 = 0. 9956. This logarithmic decay law echoes the scale of the prime number theorem pn ∼ nlnn, indicating that the logarithmic scale in Dirichlet’s theorem and the logarithmic decay law in the sieving process are inherently correlated. Through theoret- ical analysis and numerical experiments, this paper firmly roots the modular sieve method in classical number theory and provides a new perspective for understanding the dynamic laws of prime distribution.