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On the distribution of algebraic numbers with prescribed factorization properties. (English) Zbl 1136.11064
Summary: We study irregularities in the distribution of elements with prescribed factorization properties in reduced semigroups of non-zero algebraic integers and some other semigroups of ideals. In particular, we show that the counting functions of the sets of elements that are products of \(k\), or no more than \(k\), irreducibles, elements whose all factorization lengths are in a given range, and, in case the class number is greater than \(2\), elements with factorizations of at most \(m\) distinct lengths, \(m\) sufficiently large, or exactly \(m\) distinct lengths, \(m \geq 2\), oscillate around their main terms. The results for the sets \(G_m\), of elements with factorizations of at most \(m\) distinct lengths, for small \(m\) depend on the combinatorial properties of the class group or on the multiplicities of non-real zeros of Hecke zeta functions of the field considered. The properties of such multiplicities are studied for several fields using computational methods.

MSC:
11N37 Asymptotic results on arithmetic functions
11N45 Asymptotic results on counting functions for algebraic and topological structures
11R27 Units and factorization
11R42 Zeta functions and \(L\)-functions of number fields
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