2008/06/10 by Peter Borwein, Michael James Coons, Borwein, Peter +2
Computer Science · Mathematics · #11A05 #11J81 #Advanced Mathematical Identities #Analytic Number Theory Research #FOS: Mathematics #Number Theory (math.NT) #math.NT #msc:11A05 #msc:11J81 #semigroups and automata theory
paper · pdf · doi:10.48550/arxiv.0806.1694
17 pages
arxiv created 2008/06/10 · openalex publication_date 2008/06/10 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
\em The Liouville number, denoted l, is defined by l:=0.100101011101101111100..., where the nth 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 nth 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 ∑k=0^∞\frac23k23k2+23k+1=0.101100101101100100101..., where the nth 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 \citeDek1 as well as a theorem of Mahler's \citeMahl1. (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.