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A generalization of Davenport’s constant and its arithmetical applications. (English) Zbl 0760.11031
The author proves the following theorem. Let $$R$$ be the ring of integers of an algebraic number field $$K$$ and let $$G$$ denote the ideal class group of $$R$$. For every natural $$k$$ let $$M_ k(x)$$, $$[M_ k'(x)]$$ denote the number of principal ideals of $$R$$ with norms $$\leq x$$ generated by elements with all factorizations into irreducibles of lengths $$\leq k$$ [having at least one factorization of length $$\leq x$$, resp.] Then for $$x\geq e^ e$$ and $$q\in\mathbb{Z}$$, $$0\leq q\leq c_ 0{\sqrt\log x\over\log\log x}$$, we have $M_ k(x)={x\over\log x}\left[\sum^ q_{\mu=0}{W_ \mu(\log\log x)\over(\log x)^ \mu}+O\left((c_ 1q)^ q{(\log\log x)^{D_ k(G)}\over(\log x)^{q+1}}\right)\right]$ and $M_ k'(x)={x\over\log x}\left[\sum^ q_{\mu=0}{W_ \mu'(\log\log x)\over(\log x)^ \mu}+O\left((c_ 1q)^ q{(\log\log x)^{kD(G)}\over(\log x)^{q+1}}\right)\right],$ where $$c_ 0$$, $$c_ 1$$ are positive constants, and $$W_ \mu$$, $$W_ \mu'\in C[X]$$ are polynomials such that $$\deg W_ \mu\leq D_ k(G)$$, $$\deg W_ \mu'\leq kD_ 1(G)$$, $$\det W_ 0=D_ k(G)-1$$, $$\deg W_ 0'=kD_ 1(G)-1$$, and $$W_ 0$$, $$W_ 0'$$ have positive leading coefficients. $$D_ k(G)$$ denote here certain combinatorial constants depending only on $$G$$. This generalizes the results of P. Rémond [Ann. Sci. Éc. Norm. Supér., III. Ser. 83, 343-410 (1966; Zbl 0157.096)] and the reviewer [Acta Arith. 43, 53-68 (1983; Zbl 0526.12006)]. The principal novelty of the result is the introduction of a new type of combinatorial constants $$D_ k(G)$$. In case $$k=1$$, $$D_ 1(G)$$ equals to the classical Davenport’s constant [see W. Narkiewicz, Elementary and Analytic Theory of Algebraic Numbers (Springer, 1990; Zbl 0717.11045)]. The constants $$D_ k$$, $$k>1$$, form a sequence of new invariants of the group $$G$$; some basic properties of these numbers are considered in the paper as well.

##### MSC:
 11R27 Units and factorization 11R47 Other analytic theory
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