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Factorization of natural numbers in algebraic number fields. (English) Zbl 0729.11054
Quantitative aspects of factorization in algebraic number fields have been first systematically studied by W. Narkiewicz in the 60’s and then attracted attention of many authors. Counting functions of sets of algebraic integers having prescribed divisibility properties and norm \(\leq x\) have usually asymptotics of the form (x\(\to \infty)\) \(x^ A(\log x)^ B(\log \log x)^ C\) with A, B, C depending on the problem under consideration and on the basic number field K. The present author deals with the function \(G'_ m(x)=\#\{n\in {\mathbb{N}} |\) \(n\leq x\), n has at most m factorizations of distinct lengths in \(K\}\) and \(\bar F_ m(x)=\#\{(a) || N_{K/Q}(a)| \leq x\), a has m distinct factorizations in \(K\}\). \(G'_ m(x)\) has been considered before by J. Śliwa [Acta Arith. 31, 399-417 (1976; Zbl 0347.12005)] who proved that \[ G'_ m(x)\quad \sim \quad x(\log x)^{-\eta '(K,m)}(\log \log x)^{\psi '(K,m)}. \] The present author gives explicit expressions for the exponents \(\eta '(K,m)\) and \(\psi '(K,m)\). The most interesting fact here is that \(\eta '(K,m)=\eta '(K)\) is independent of m. Using a general result of the reviewer [Acta Arith. 43, 53-68 (1983; Zbl 0526.12006)] he also proves that \[ \bar F_ m(x)=x(\log x)^{-1+1/h} \bar W_ m(\log \log x)+O(x(\log x)^{-2+t/h}(\log \log x)^{\tau_ m}), \] where \(\bar W_ m\in {\mathbb{C}}[X]\), \(\bar C_ m\geq 0\). This improves a result by W. Narkiewicz [Acta Arith. 21, 313-322 (1972; Zbl 0215.073)].

MSC:
11R27 Units and factorization
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