Independence measures of arithmetic functions

Abstract

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

Applications of differential algebra to algebraic independence of arithmetic functions

“…Proposition 4.1 generalizes Proposition 2.1 of [9]. For example, by taking D = ∂L, one sees that C is algebraically closed in A and that Log(f ), f, Exp(f ) are algebraically independent over C for f ∈ A+ \C.…”

“…. , fn over C. Several results about this measure were proved in [9]. Our method, due to its non-constructive nature, cannot produce those results.…”

“…By setting the m in Proposition 4.6 to various values, one obtains strengthened versions of Test I-IV in [9]. These tests were used to establish algebraic independence of various Fibonacci and Lucas zeta functions [9, Proposition 2.5, 2.6].…”

“…Our method, due to its non-constructive nature, cannot produce those results. However, the non-quantitative part of both Theorem 3.2 and Theorem 3.4 of [9] can be generalized as follows. Proof.…”

On the Sum of Reciprocal Generalized Fibonacci Numbers

“…[1], [3], [6], and [11]). Such reciprocal sums of Fibonacci-type numbers have been studied by several authors (e.g.…”