What question did this study set out to answer?

This research aims to explore the decay pattern of candidate set size in the modular sieve method and its relationship to the prime number theorem.

March 10, 2026Open Access

The Logarithmic Law of Candidate Set Decay in the Modular Sieve Method and Its Connection to the Prime Number Theorem

Key Points

This research aims to explore the decay pattern of candidate set size in the modular sieve method and its relationship to the prime number theorem.
Analyzed data from 47 sieving steps using logarithmic fitting
Examined candidate set size S(n) against sieving step n
Applied the model S(n) = A − B ln n
Assessed goodness of fit R2 values for ranges N = 1000000 and N = 10000000
Identified a high goodness of fit (R2 = 0.995555) for S(n) model
Confirmed the universality of the decay pattern across different ranges (R2 = 0.9964)
Established a connection between candidate set decay and the prime number theorem’s logarithmic scale

Abstract

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.

Read Full Paperexternally

AI से पूछें

Bookmark

View Full Paper