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1. For an additively written finite abelian group G, Davenport’s constant D(G) is defined as the maximal length d of a sequence (g1, . . . , gd) in G such that ∑d j=1 gj = 0, and ∑ j∈J gj 6= 0 for all ∅ 6= J {1, . . . , d}. It has the following arithmetical meaning: Let K be an algebraic number field, R its ring of integers and G the ideal class group of R. Then D(G) is the maximal number of prime ideals (counted with multiplicity) which can divide an irreducible element of R. This fact was first observed by H. Davenport (1966) and worked out by W. Narkiewicz [8] and A. Geroldinger [4]. For a subset Z ⊂ R and x > 1 we denote by Z(x) the number of principal ideals (α) of R with α ∈ Z and (R : (α)) ≤ x. If M denotes the set of irreducible integers of R, then it was proved by P. Rémond [12] that, as x →∞, M(x) ∼ Cx(log x)−1(log log x)D(G)−1 ,

A generalization of Davenport's constant and its arithmetical applications