Observations About Algebraic Dependence of Dirichlet Series

Applications of differential algebra to algebraic independence of arithmetic functions

“…Our next result generalizes both [19,Theorem 3.3] and [16, Theorem 3] by relaxing the assumption that supp f contains infinitely many primes to that supp f is not finitely generated. The proof below is a mixture of the those given in [19] and [16]. Therefore, our sole contribution here is the realization that these proofs remain valid in a more general setting.…”

“…However, to the best of our knowledge, all existing applications of this criterion involve only the log-derivation and the basic derivations so to all of them Theorem 3.5 is applicable. In the next two sections, we will generalize a number of results in [19,9] and [16] in various directions.…”

“…The lemma follows since fi(mi) is non-zero for each i. Lemma 4.9 was used to prove Theorem 7 in [16]. It states that a set of nonzero non-invertible arithmetic functions is algebraically independent over C if the norms of its members are pairwise relatively prime.…”

“…The converse is also true and it follows easily from the fact that em * en = emn for any n, m ∈ N. Thus for a set of natural numbers N , the necessary and sufficient condition for Exp * {en : n ∈ N } to be algebraically independent over C is that the elements of N are multiplicatively independent. Note that Theorem 7 in [16] alone does not imply this fact since there are multiplicatively independent numbers such as 2 and 6 that are not relatively prime.…”

“…The central result about algebraic independence of arithmetic functions is the following criterion of Shapiro and Sparer [19,Theorem 3.1]. We refer the reader to [19,9] and [16] for its numerous applications.…”