(Non)Automaticity of number theoretic functions

TL;DR

This paper proves that Liouville's function and several other number-theoretic functions are not automatic and their generating functions are transcendental over finite fields, using classical prime distribution results.

Contribution

It establishes the non-automaticity of Liouville's function and related functions, and shows their generating functions are transcendental over finite fields.

Findings

Liouville's function is not k-automatic for any k>2.

The generating series of these functions are transcendental over _p(X).

Results apply to multiple classical number-theoretic functions.

Abstract

Denote by $\lambda(n)$ Liouville's function concerning the parity of the number of prime divisors of $n$. Using a theorem of Allouche, Mend\`es France, and Peyri\`ere and many classical results from the theory of the distribution of prime numbers, we prove that $\lambda(n)$ is not $k$--automatic for any $k> 2$. This yields that $\sum_{n=1}^\infty \lambda(n) X^n\in\mathbb{F}_p[[X]]$ is transcendental over $\mathbb{F}_p(X)$ for any prime $p>2$. Similar results are proven (or reproven) for many common number--theoretic functions, including $\phi$, $\mu$, $\Omega$, $\omega$, $\rho$, and others.