A GENERALIZATION OF DAVENPORT’S CONSTANT AND ITS ARITHMETICAL APPLICATIONS BY

1. For an additively written finite abelian groupG, 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, where C> 0 depends on K; the error term in this asymptotic formula was investigated by J. Kaczorowski [7]. If an element α ∈ R \ (R × ∪{0}) has a factorization α = u1 ·... · ur into irreducible elements uj ∈ R, we call r the length of that factorization and denote by L(α) the set of all lengths of factorizations of α. For k ≥ 1, we define sets Mk and M ′k (depending on K) as follows: Mk consists of all α ∈ R \ (R × ∪ {0}) for which maxL(α) ≤ k (i.e., α has no factorization of length r> k); M ′k consists of all α ∈ R \ (R×∪{0}) for which minL(α) ≤ k (i.e., α has a factorization of length r ≤ k). If G = {0}, then Mk = M ′k for all k; in the general case, we have M1 =M ′1 =M and Mk ⊂M ′k for all k. In this paper, we generalize the results of Rémond and Kaczorowski and obtain asymptotic formulas for Mk(x) and M ′k(x). To do this, we shall define a sequence of combinatorial constants Dk(G) (k ≥ 1) generalizing D(G) = D1(G), and we shall obtain the following result. 204 F. HALTER-KOCH Theorem. For x ≥ ee and q ∈ Z, 0 ≤ q ≤ c0 log x log log x, we have Mk(x) = x log x [ q∑ µ=0 Wµ(log log x) (log x)