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arXiv · 0810.3709

(Non)Automaticity of number theoretic functions

Michael Coons
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Abstract

Denote by $λ(n)$ Liouville's function concerning the parity of the number of prime divisors of $n$. 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 $λ(n)$ is not $k$--automatic for any $k> 2$. This yields that $\sum_{n=1}^\infty λ(n) X^n\in\mathbb{F}_p[[X]]$ 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 $ϕ$, $μ$, $Ω$, $ω$, $ρ$, and others.

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Michael Coons. 2008-10-29. (Non)Automaticity of number theoretic functions. https://arxiv.org/abs/0810.3709

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