On the density of cyclic quartic fields (Q2739256)

Let \(h(n)\) denote the number of cyclic quartic fields over the rational number field \(Q\) with discriminant \(n\) and set \(N(x)= \sum_{n\leq x}h(n)\). Then, \textit{A. M. Baily} provided an asymptotic formula for \(N(X)\) [cf. J. Reine Angew. Math. 315, 190-210 (1980; Zbl 0421.12007)]. However, his generating function \(f(s)= \sum_{n=1}^\infty \frac{h(n)}{n^s}\) is given incorrectly. In this paper, using the representation of a cyclic quartic field given by one of the authors and others and an elementary method, they correct Baily's result.