Transcendence of the Gaussian Liouville number and relatives

TL;DR

This paper proves the transcendence of the Gaussian Liouville number and related numbers, extending the known properties of the classical Liouville number using methods involving formal power series and multiplicative functions.

Contribution

It introduces the transcendence of the Gaussian Liouville number and similar constructs, employing methods inspired by Dekking and Mahler's theorems.

Findings

Gaussian Liouville number is transcendental

Related numbers defined by prime divisor parity are transcendental

Methods involve generating functions of multiplicative functions

Abstract

{\em The Liouville number}, denoted $l$, is defined by $$l:=0.100101011101101111100...,$$ where the $n$th bit is given by ${1/2}(1+\gl(n))$; here $\gl$ is the Liouville function for the parity of prime divisors of $n$. Presumably the Liouville number is transcendental, though at present, a proof is unattainable. Similarly, define {\em the Gaussian Liouville number} by $$\gamma:=0.110110011100100111011...$$ where the $n$th bit reflects the parity of the number of rational Gaussian primes dividing $n$, 1 for even and 0 for odd. In this paper, we prove that the Gaussian Liouville number and its relatives are transcendental. One such relative is the number $$\sum_{k=0}^\infty\frac{2^{3^k}}{2^{3^k2}+2^{3^k}+1}=0.101100101101100100101...,$$ where the $n$th bit is determined by the parity of the number of prime divisors that are equivalent to 2 modulo 3. We use methods similar to that of Dekking's proof of the transcendence of the Thue--Morse number \cite{Dek1} as well as a theorem of Mahler's \cite{Mahl1}. (For completeness we provide proofs of all needed results.) This method involves proving the transcendence of formal power series arising as generating functions of completely multiplicative functions.