In this note, we show that if b > 1 is an integer, f(X) ∈ Q[X] is an integer valued quadratic polynomial and K > 0 is any constant, then the b-adic number ∑n≥0 (an / bf(n)), where an ∈ Z and 1 ≤ |an| ≤ K for all n ≥ 0, is neither rational nor quadratic.
Denote by Liouville’s function concerning the parity of the number of prime divisors of . Using a theorem of Allouche, Mendès France, and Peyrière and many classical results from the theory of the distribution of prime numbers, we prove that is not –automatic for any . This yields that is transcendental over for any prime . Similar results are proven (or reproven) for many common number–theoretic functions, including , , , , , and others.