Independence measures of arithmetic functions (Q607032)

Consider the ring of arithmetic functions with the usual addition and the convolution as multiplication, isomorphic to the ring of formal Dirichlet series over the complex numbers. Concerning algebraic independence of these functions, there is a beautiful criterion of \textit{H. N. Shapiro} and \textit{G. H. Sparer} only involving the values of certain derivatives of the arithmetic functions on certain primes [Commun. Pure Appl. Math. 39, 695--745 (1986; Zbl 0605.30003)]. The authors of the present paper extend this result deriving measures of algebraic independence and give some nice examples.