Explicit formulas of Delange-Selberg type for some classes of arithmetic functions on the semigroup \(G_{\mathcal K}\)
Let \(K\) be an algebraic number field and let functions \(f\) and \(F\) be defined on the set of all non-zero integral ideals of \(K\). Let \(f\) be multiplicative, complex valued and bounded and let \(F\) be additive and take values in the set \(\mathbb{N}\cup \{0\}\). Suppose also that for every prime ideal \({\mathcal P}\) we have \(f({\mathcal P})= F({\mathcal P})=1\). Then for every complex \(| z|\leq 1\) and real \(x\geq x_ 0(q,K)\) the following formula holds \[ \sum_{N(I)\leq x} f(I) z^{F(I)}= x\sum_{j=0}^ q A_ j(z) (\log x)^{z-j-1}+ O(C(q,K) x(\log x)^{\text{Re }z-q-2}) \] with certain explicitly given functions \(A_ j\) which are continuous in the disc \(| z|\leq 1\) and holomorphic in its interior. The author estimates \(x_ 0(q,K)\) as well as \(C(q,K)\) in terms of \(q\), the degree and the discriminant of \(K\). In the rational case a similar theorem has been proved by \textit{H. Delange} [Acta Arith. 19, 105-146 (1971; Zbl 0217.319)]. The author concludes the paper by giving a number of corollaries dealing with some special cases of the main result \((\sum_{N(I)\leq x} z^{\omega(I)}\), \(\sum_{N(I)\leq x} z^{\Omega(I)}\), etc.). [Reviewer's remark: In the author's expressions for \(x_ 0(q,K)\) and \(C(q,K)\) the dependence on \(R:= \sup_ I | f(I)|\) is missing. So the constants \(C_{19}\) and \(C_{20}\) in the theorem are not numerical, but may depend on \(R\)].
scientific article; zbMATH DE number 556370