Numerical Coincidences between Prime Proportions from a Modular Sieve and Non-trivial Zeros of the Riemann ζ Function

This paper reports an unexpected observation derived from a modular sieve method for prime generation: the proportions of primes obtained under different moduli products systematically coincide with the imaginary parts of the first few non-trivial zeros of the Riemann ζ function. The method is based on an elementary number-theoretic fact: all primes greater than 5 fall into eight residue classes mod- ulo 30 (the “8-orbit” structure). By counting residue frequencies and applying a dynamic threshold, we obtain a series of prime proportions. At the scale N = 106, the deviations between 16 proportions and the zeros #6 to #14 are mostly within 0.2, with one point as small as 0.00345. Using these proportions to synthesize a wave and performing statistical tests on prime positions, we find: for x ≤ 100, 20 out of 25 primes (80.0%) lie on the positive side of the wave (p = 0.0016); for x ≤ 500, 63 out of 95 primes (66.3%) lie on the positive side (p = 7.6 × 10−4); for x ≤ 1000, 100 out of 168 primes (59.5%) lie on the positive side (p = 0.005). As x increases, the positive proportion gradually converges to 50%, in agreement with the amplitude decay x1/2/ ln x predicted by Riemann’s explicit formula. Repeating the test at x ≤ 1000 with a different set of 34 proportions obtained at N = 107 still yields 56.0% positive (p = 0.06), confirming the robustness of the phenomenon. The convergence coefficient δ is locked to 0.5 or 1.0 after optimization, numerically coinciding with the critical line R(s) = 1/2 of the Riemann hypothesis. These ob- servations offer a new empirical perspective on the connection between Riemann zeros and the distribution of primes.