On a problem of R. Brauer on zeta-functions of algebraic number fields
Let \(\zeta_ X(s)\) denote the \(\zeta\)-function of Dedekind on an algebraic number field X over \({\mathbb{Q}}\), with X of finite degree over \({\mathbb{Q}}\). \textit{R. Brauer} [Acta Arith. 24, 325-327 (1973; Zbl 0272.12011)] showed that \(\zeta_ K(s)\zeta_ k(s)/\zeta_{K_ 1}(s)\zeta_{K_ 2}(s)\) is an entire function, whenever \(K_ 1\) and \(K_ 2\) are algebraic number fields, both normal over their intersection k and with compositum \(K=<K_ 1,K_ 2>.\) Here it is proved that this quotient of \(\zeta\)-functions is also entire, if (1) \(K_ 1={\mathbb{Q}}(^ p\sqrt{n})\), \(K_ 2={\mathbb{Q}}(^ p\sqrt{m})\), \((n,m)=1\), n and m p-free integers, p odd prime or if (2) \(K_ 1={\mathbb{Q}}(^ p\sqrt{n})\), \(K_ 2={\mathbb{Q}}(^ q\sqrt{m})\), p and q odd distinct primes, n some p-free integer and m a q-free integer, respectively. [Reviewer's remark: a positive integer t is said to be p-free here, if for every prime r, \(r^ p\) does not divide t.]
scientific article; zbMATH DE number 3943968