Independence measures of arithmetic functions II

Abstract

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-

Applications of differential algebra to algebraic independence of arithmetic functions

“…Incidentally, this shows that Theorem 2.2 of [10] is not true. Moreover, ∂Lf1(n) = k log(2) if n = 2 k for some k ≥ 0; 0 otherwise.…”

“…For example, none of the term in the null sequence (ep) is even in A 2 0 because members of A 2 0 vanish on every prime. Our second observation is about linear independence of arithmetic functions over C. It was proved [10,] that arithmetic functions f1, . .…”

“…This falsifies Corollary 2.3-2.5 of [10]. In particular, it is not true that a Dirichlet series which is not a Dirichlet polynomial is hyper-transcendental over C. Corollary 2.6 and 2.7 of [10] are also problematic.…”

“…Our last few remarks are about Theorem 3.5 and Section 2 of [10]. In searching for a generalization of the Jacobian criteria, we realize that the derivations in Theorem 3.5 cannot be replaced by differential operators.…”

“…Viewing the linear independence result of [10] as one about differential fields frees us from focusing on the log-derivation: if the Wronskian of f1, . .…”