Partition-theoretic model of prime distribution

TL;DR

This paper introduces a novel deterministic model based on partition theory that predicts prime distribution and related conjectures, providing accurate estimates of prime counts and insights into prime gaps.

Contribution

The paper applies partition theory to model prime distribution, deriving predictions consistent with known theorems and offering improved numerical estimates of prime counts.

Findings

Model predicts prime number theorem and twin prime conjecture.

Provides near-exact estimates of prime counts up to 10,000.

Limited evidence of predictable prime gap variations.

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 $n\geq 2$, $$p_{n}\ =\ 1\ +\ 2\sum_{j=1}^{n-1}\left\lceil \frac{d(j)}{2}\right\rceil\ +\ \varepsilon(n),$$ where $p_k$ is the $k$th prime number, $d(k)$ is the divisor function, and $\varepsilon(k)$ 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 $\pi(n)$ similar to those provided by the logarithmic integral, and much more accurate than $\operatorname{li}(n)$ up to $n=10{,}000$ where the estimates are {\it almost exact}. We then perform computational tests of unusual predictions of the model, finding limited evidence of predictable variations in prime gaps.