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Partition-theoretic model of prime distribution

Aidan Botkin, Madeline L. Dawsey, David J. Hemmer, Matthew R. Just, Robert Schneider

TL;DR

This work introduces a deterministic, partition-theoretic model of prime distribution centered on the formula $p_n = 1 + 2 \sum_{j=1}^{n-1} \left\lceil \frac{d(j)}{2}\right\rceil + \varepsilon(n)$, with $d$ the divisor function and $\varepsilon(n)$ negligible, bridging partition theory and multiplicative number theory via norm and supernorm statistics. The authors demonstrate that Model 1 naturally yields the prime-number theorem regime $p_n \sim n\log n$ and posits a mechanism for twin primes through the case $d(n)=2$, while also predicting local prime-gap variations; they subsequently refine the model (Model 2, Model 2*) by incorporating $\pi_2(p_{n-1})$ corrections to produce highly accurate estimates for $\pi(n)$ up to $n=10^4$ and beyond. Extensive computational testing validates qualitative alignment with known prime-distribution patterns and reveals both promising predictions and notable limitations, particularly for larger $n$ and for twin-prime behavior. The work presents a complementary, highly structured framework to the probabilistic models of primes, offering new avenues for testing partition-theoretic influences on prime gaps and local fluctuations.

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 $π(n)$ 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.

Partition-theoretic model of prime distribution

TL;DR

This work introduces a deterministic, partition-theoretic model of prime distribution centered on the formula , with the divisor function and negligible, bridging partition theory and multiplicative number theory via norm and supernorm statistics. The authors demonstrate that Model 1 naturally yields the prime-number theorem regime and posits a mechanism for twin primes through the case , while also predicting local prime-gap variations; they subsequently refine the model (Model 2, Model 2*) by incorporating corrections to produce highly accurate estimates for up to and beyond. Extensive computational testing validates qualitative alignment with known prime-distribution patterns and reveals both promising predictions and notable limitations, particularly for larger and for twin-prime behavior. The work presents a complementary, highly structured framework to the probabilistic models of primes, offering new avenues for testing partition-theoretic influences on prime gaps and local fluctuations.

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 , where is the th prime number, is the divisor function, and 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 similar to those provided by the logarithmic integral, and much more accurate than up to 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.
Paper Structure (24 sections, 3 theorems, 43 equations, 4 figures, 3 tables)

This paper contains 24 sections, 3 theorems, 43 equations, 4 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Model of primes from partition first principles
  3. Concepts and notations
  4. Multiplicative partition statistics and a multiplicative theory of (additive) partitions
  5. Isomorphism between partitions and natural numbers
  6. Comparing the norm and supernorm
  7. Partition model of prime distribution
  8. Partition-theoretic model of prime gaps
  9. Simplifying assumption
  10. Initial model
  11. Predictions from Model 1
  12. Critique of Model 1
  13. Testing the model
  14. Testing predictions from Model 1
  15. Methods
  16. ...and 9 more sections

Key Result

Theorem 1

The supernorm map $\widehat{{N}}\colon \mathcal{P} \to \mathbb Z^+$ is an isomorphism between monoid $(\mathcal{P}, \ \cdot\ )$ with partition multiplication and monoid $(\mathbb Z^+, \ \cdot\ )$ with integer multiplication.

Figures (4)

  • Figure 1: Plot of $P(n)$ up to $n=100{,}000{,}000$
  • Figure 2: Relative error in the Model 1 estimate for $p_n$ up to $n=100{,}000$
  • Figure 3: Relative error in the Model 2 estimate for $p_n$ up to $n=100{,}000$
  • Figure 4: Relative error in the Model 2* estimate for $p_n$ up to $n=100{,}000$

Theorems & Definitions (19)

  • Definition 1.1
  • Definition 1.2
  • Definition 1.3
  • Theorem : Dawsey--Just--Schneider
  • Proposition 1.4
  • proof
  • Proposition 2.1
  • proof
  • Remark 2.1
  • Remark 2.2
  • ...and 9 more