On the Asymptotics of a Prime Spin Relation

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$(\mathfrak{p},σ)$spin$(\mathfrak{p},σ^{-1})=1$ for all $σ\neq 1 \in$ 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.