The notion of algebraic dependence in the ring of arithmetic functions with addition and Dirichlet product is considered. Measures for algebraic independence are derived.

Independence measures of arithmetic functions II by Takao Komatsu (Hirosaki), Vichian Laohakosol (Bangkok) and Pattira Ruengsinsub (Bangkok)1. Introduction. In our earlier work, the notion of independence measure of arithmetic functions was introduced and two main results ([3, Theorems 3.2 and 3.4]) about such measure were proved. These results are proved under the hypothesis that there is a set of distinct primes for which the set of vectors of function values at points depending on these primes is linearly independent over C, and the proofs make use of the first assertion of [3, Lemma 3.3] where the p-basic derivation is the main tool. Our first objective here is to improve upon these results by replacing the set of primes by any set of distinct natural numbers enjoying similar properties. This is accomplished by making use of the second assertion of [3, Lemma 3.3] where the log-derivation is employed instead.To systematize our presentation, we first recall all relevant terminology. Denote by (A, +, * ) the unique factorization domain of arithmetic functions equipped with addition and convolution (or Dirichlet product) defined byand writeThe convolution identity, I, is defined by I(1) = 1 and I(n) = 0 for all n > 1. An arithmetic function f is called a unit (in A) if its convolution inverse f −1 exists, and this is the case if and only if f (1) = 0. It is well-known, [8, Chapter 4], that (A, +, * ) is isomorphic to (D, +, •), whereis the ring of formal Dirichlet series equipped with addition and multipli-

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