Partition-theoretic model of prime distribution, II
Robert Schneider
TL;DR
The paper investigates a deterministic, partition-theoretic framework for prime distribution, aiming to estimate $\pi(n)$ with a sequence of increasingly refined models. It introduces partition-based statistics such as the norm $N(\lambda)$ and the supernorm $\widehat{N}(\lambda)$, leading to a core formula for the $n$th prime $p_n = 1 + 2\sum_{j=1}^{n-1} \lceil d(j)/2\rceil + \varepsilon(n)$, where the error term $\varepsilon(n)$ differentiates Model 1, Model 2, and Model 2*. The authors then develop Model 3 by incorporating two tunable parameters, $r$ and $t$, to interpolate between the previous models and improve estimates of $\pi(n)$; their computational experiments show that with $(r,t)=(6,0.11)$ the model can closely match $\pi(n)$ up to $n$ around $10^6$, while the resulting $p_n$ sequence may diverge from the true primes in later terms. These results illustrate that partition theory encodes information about primes and offers a discrete, structure-driven perspective on prime distribution, potentially complementing classical probabilistic and analytic approaches.
Abstract
In recent work by Botkin, Dawsey, Hemmer, Just and the present author, a deterministic model of prime number distribution is developed based on properties of integer partitions that gives almost exact estimates for $π(n)$, the number of primes less than or equal to positive integer $n$, up to $n=10{,}000$. In this follow-up paper, the author summarizes the ideas behind this partition-theoretic model of primes and formulates a computational model that is practically exact in its estimates of $π(n)$ up to $n=100,000$.