This paper reports on the decay pattern of the total candidate set size during the sieving process of the modular sieve method. Through logarithmic fitting of data from 47 sieving steps, starting from the first prime product modulus 7 × 11 within the range N = 10000000, I discovered that the candidate set size S(n) as a function of the sieving step n follows the model S(n) = A − B ln n with exceptional precision, achieving a goodness of fit R2 = 0.995555. This pattern holds true for the range N = 1000000 as well (R2 = 0.9964), indicating its universality. This logarithmic decay is intrinsically linked to the logarithmic scale found in the Prime Number Theo- rem, where the n-th prime pn ∼ nlnn, revealing a profound connection between the sieving step count and the natural logarithm. This work provides a precise mathematical description for understanding the statistical properties of the sieve process.
Huang Feiyue (Sun,) studied this question.