Evaluation of the Dedekind zeta functions at \(s=-1\) of the simplest quartic fields
The author considers cyclic quartic fields \(K_t=\mathbb Q(\theta_t)\), \(\theta_t\) being a zero of \(X^4-tX^3-6X^2+tX+1\), where \(t\) is a positive integer such that \(t^2+16\) has no odd square factor (such fields are called simple quartic fields). \textit{C. L. Siegel} [Nachr. Akad. Wiss. Gött., II. Math.-Phys. Kl. 1969, 87--102 (1969; Zbl 0186.08804)] gave a formula for the value of the Dedekind zeta-function \(\zeta_K(s)\) of totally real fields \(K\) at negative rational integers, which for quartic fields implies \[ \zeta_K(-1)={1\over30}\sum_a\sum_{I|a\delta_K}N(I), \] where \(\delta_K\) is the different of \(K\), and \(a\) runs over totally positive elements of \(\delta_K^{-1}\) of trace \(1\). The author presents a geometric method of calculating the right hand-side of this formula, and presents a table of \(\zeta_{K_t}(-1)\) for \(t=1,2,\dots,21\).
scientific article; zbMATH DE number 6338993