A limit formula for \(\zeta (2k+1)\)
The author studies the Riemann zeta-function \(\zeta(s)\) at positive odd integers \(s=n=2k+1\). Let \(q\) be a prime \(\equiv 1 \pmod n\) and let \(K\) be the subfield of degree \(n\) of the \(q\)-th cyclotomic field. Denote by \(I(K)\) the set of all non-zero integral ideals in \(K\) and by \(\mathcal O_{K,0}\) the \(\mathbb{Z}\)-module consisting of the integers in \(K\) with vanishing trace. The main result of the paper is the formula \[ \zeta(2k+1) = \frac{2q^n-q^{n-1}-q}{q^n-1}\zeta(2k) \lim_{x\to\infty}\frac{j_K(x)}{j_{K,0}(x)}, \] where \(j_K(x)\) is the number of \(A \in I(K)\) with Norm\((A)\leq x\) and \(j_{K,0}(x)\) is the number of \(A \in I(K)\) such that the group \(\mathcal O_{K,0}/(\mathcal O_{K,0} \cap A)\) is of order \(\leq x\). The function \(j_{K,0}(x)\) was introduced and studied by the author in an earlier work [Proc. Intern. Conf. Pure Appl. Math., Beijing, 1994, 141-156; Chin. Sci. Bull. 41, No. 8, 701-702 (1996; Zbl 0876.11051)]. Motivated by the above formula the author now goes on to discuss elementary properties of the function \(j_K(x)/j_{K,0}(x)\), assuming that \(K\) is an algebraic number field subject to various conditions.
scientific article; zbMATH DE number 1375091