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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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