A Statistical Study of Prime Distribution Based on a Modulo 30 Sieve

This paper starts from an elementary number-theoretic fact: all primes greater than 5 can only have 8 possible residue classes mod- ulo 30. Based on this structure, we classify and count primes, and introduce a statistical sieve method based on residue frequencies. In the range up to 1010, we find that the ratio of prime pairs with gap 30 to twin primes is 2.66646, which is in excellent agreement with the Hardy–Littlewood conjecture’s theoretical value 8/3 (relative er- ror < 0.01%). Furthermore, we apply stepwise sieving in the ranges 107 and 108, stabilizing the proportion of primes in the candidate set near 50%. By verifying the 8 orbits separately, the prime proportions in each residue class lie between 49.8% and 50.1%, with an average of 49.96%, fully demonstrating the uniformity and convergence of the sieve method. The convergence coefficient 0.5 that appears in the statistical sieve is revealed as an intrinsic constant of the system; its numerical coincidence with the critical line 1/2 of the Riemann ζfunction warrants further investigation.