Counting algebraic units with bounded height (Q687520)

Let \(\mathbb K\) be a number field of degree \(d\) whose unit group \(U_{\mathbb K}\) has rank \(r\geq 2\). Let \(\sigma_ i: \mathbb K\to\mathbb C\) denote the \(d\) distinct embeddings of \(\mathbb K\) into \(\mathbb C\). For \(u\in U_ {\mathbb K}\) let \(H(u)= \max_{1\leq i\leq d} |\sigma_ i(u)|\) define its height. Finally let \(R\) denote the regulator of \(U_{\mathbb K}\). Let \(q\) be a positive real number and set \(U(q)= |\{u\in U_ {\mathbb K}: H(u)<q\}|\). Then it is proved in this paper that there is a positive constant \(c=c(d)\), depending only on \(d\), such that \[ U(q)= {{\omega(r+1)^ r} \over {Rr!}} (\log(q))^ r+ O((\log(q))^{r-1- (cR^{2/r})^{-1}}) \] as \(q\to\infty\). One can take \(c=6.2\cdot 10^{12} d^{10} (1+2\log(d))\). This result considerably improves the estimate of the error term proved by the first author [J. Aust. Math. Soc., Ser. A 53, 39--50 (1992; Zbl 0765.11028)]. It should be remarked that equation (37) of the cited paper is now corrected [J. Aust. Math. Soc., Ser. A 56, No. 1, 144 (1994; Zbl 0806.11031)] and so the earlier and the present results are consistent.