Partition-theoretic model of prime distribution

Document Type

Article

Publication Date

7-1-2026

Abstract

We make an application of ideas from partition theory to a problem in multiplicative number theory. We propose a deterministic model of prime number distribution, from first principles related to properties of integer partitions, that naturally predicts the prime number theorem as well as the twin prime conjecture. The model posits that, for n≥2, pn=1+2∑j=1n−1⌈d(j)2⌉+ε(n), where pk is the k th prime number, d(k) is the divisor function, and ε(k) is an explicit error term that is negligible asymptotically; both the main term and error term represent enumerative functions in our conceptual model. We refine the error term to give numerical estimates of π(n) similar to those provided by the logarithmic integral, and much more accurate than li(n) up to n=10,000 where the estimates are almost exact . We then perform computational tests of unusual predictions of the model, finding limited evidence of predictable variations in prime gaps.

Publication Title

Journal of Number Theory

Link to Full Text

Share

COinS