Robert.Schneider

Affiliation: Michigan Technological University

Title Of Talk: Partition-theoretic model of prime distribution

Abstract: 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. 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 $n=100{,}000$ where the estimates are almost exact.

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