Distribution of units of a cubic abelian field modulo prime numbers (Q2497094)

The author studies the (multiplicative) group \(E(\mathfrak n)\) of those residue classes modulo an ideal \(\mathfrak n\) of an algebraic number field \(F\), which contain elements of \(\mathfrak o_F^\times\), the unit group of \(F\). In this paper, \(F\) is a cyclic cubic extension of \(\mathbb Q\), and \(\mathfrak n = (p)\) the ideal generated by a rational prime \(p \in \mathbb P\), which splits in \(F\). Then obviously \(\# E((p))\) divides \(2(p-1)^2\). For \(x > 0\) put \[ T_x = \{ p \in \mathbb P \mid p \geq 3,\;p \text{ splits in } F \text{ and } \# E((p)) = 2(p-1)^2 \}. \] Using Moebius inversion, an explicit formula for \(T_x\) is given in Theorem 1, involving the Frobenius automorphisms of \(p\) for the fields \(F(\root n \of {\mathfrak o_F^\times})\) with \(n \mid (p-1)\). Based on Chebotarev's density theorem, the author conjectures that \[ \lim_{x \to \infty} \frac {\# T_x} {\text{Li} (x)} = \kappa, \] where \(\kappa\) is explicitly given, and then main part of this paper contains a proof that \(\kappa > 0\).