Transcendence of the Gaussian Liouville number and relatives

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dc.creatorBorwein, Peter
dc.creatorCoons, Michael
dc.date2008-06-10
dc.date.accessioned2026-07-07T09:43:37Z
dc.date.available2026-07-07T09:43:37Z
dc.description{\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 $$γ:=0.110110011100100111011...$$ 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.
dc.description17 pages
dc.identifierhttps://arxiv.org/abs/0806.1694
dc.identifierhttp://arxiv.org/abs/0806.1694
dc.identifier.urihttp://salesiana.dossiersoluciones.com/handle/123456789/162626
dc.subjectNumber Theory
dc.subject11J81; 11A05
dc.titleTranscendence of the Gaussian Liouville number and relatives
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