Transcendence of the Gaussian Liouville Number and Relatives

The Liouville number, denoted l, is defined by l := 0.100101011101101111100 . . . , where the nth bit is given by 1 2 (1 + λ(n)); here λ 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 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 ∞ X k=0 23 k 232 + 23 + 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 [7] as well as a theorem of Mahler’s [16]. (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. The Liouville function is the unique completely multiplicative function λ with the property that for each prime p, λ(p) = −1. Denote the sequence of λ values by L. Recall that a binary sequence is simply normal if each bit occurs with asymptotic frequency 1 2 , and normal to base 2 if each possible block of length k occurs with asymptotic frequency 2−k. The prime number theorem is equivalent to the simple normality of L; it is believed that L is normal to base 2, though a proof of normality is at present unattainable. We can get at some properties of L, though the sequence definitions of these are somewhat cumbersome. Define the Liouville number l as l := ∑ n∈N ( 1 + λ(n) 2 ) 1 2n = 0.100101001100011100001 . . . . Date: June 9, 2008. 1991 Mathematics Subject Classification. Primary 11J81; 11A05 Secondary 39B32; 11N64.