A relative class number formula for an imaginary abelian number field by means of Dedekind sum (Q1849782)

In 1955 \textit{L. Carlitz} and \textit{F. R. Olson} [Proc. Am. Math. Soc. 6, 265--269 (1955; Zbl 0065.02703)] proved a relative class number formula for the \(p\)th cyclotomic field \(\mathbb{Q} (\zeta_p)\), \(p\) an odd prime, by means of a Dedekind sum: \[ \det\bigl(s (uv^{-1},p)\bigr)_{u,v=1,2,\dots,(p-1)/2}= 2^{(p-5)/2}p^{-2} (h^*_{\mathbb{Q}(\zeta_p)})^2,\tag{1} \] where \(h^*_{\mathbb{Q}(\zeta_p)}\) is the relative class number of \(\mathbb{Q}(\zeta_p)\), \(v^{-1}\) an integer such that \(vv^{-1}\equiv 1\pmod p\) and \[ s(uv^{-1},p)=\sum^{p-1}_{k=1} \left(\frac {uv^{-1}k}{p}- \left[\frac{uv^{-1}k} {p}\right]- \frac 12 \right) \] being the Dedekind sum, \([x]\) the integral part of a rational number \(x\). They proved this formula (1) by using the formulas of Eisenstein and of Rademacher. In this note the author gives such a formula for an imaginary abelian number field. As a corollary he gets Carlitz and Olson's formula (1). The formula is deduced from his earlier paper [\textit{M. Hirabayashi}, Acta Arith. 83, 391--397 (1998; Zbl 0895.11045)]. The proof is elementary and depends only on determinant calculation.