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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.

11R27 Units and factorization
11R47 Other analytic theory
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