On the Asymptotics of a Prime Spin Relation

TL;DR

This paper derives a formula for the density of primes satisfying a specific spin relation in certain cyclic totally real number fields, with a correction issued for the inert case, but the main results remain robust.

Contribution

It provides a new explicit formula for the density of primes satisfying a spin relation in particular number fields, correcting previous errors and confirming the formula's validity.

Findings

Derived the density formula D_K for primes in specified fields

Corrected the inert case error in the original formula

Confirmed the robustness of the main results despite the correction

Abstract

For cyclic totally real number fields $K$ with odd prime degree $n$, odd class number, $2$ inert, and the property that every totally positive unit is a square, the density of rational primes $p$ that satisfy the spin relation spin$(\mathfrak{p},\sigma)$spin$(\mathfrak{p},\sigma^{-1})=1$ for all $\sigma\neq 1 \in$ Gal$(K/\mathbb{Q})$ where $\mathfrak{p}$ is a prime of $K$ above $p$ is given by the formula \[ D_K=\frac{m_Kn+1}{n2^n} \] where $m_K$ is a computable and bounded invariant of the number field $K$. This formula is modified in the erratum from the original version due to an error in the inert case. As the inert case is insubstantial, the strength of the results is not significantly changed.