Atlas: Transcendence of various numbers and series by Michael Coons

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Canadian Number Theory Association X Meeting (CNTA X)
July 13-18, 2008
University of Waterloo
Waterloo, Ontario, Canada

Organizers
Kevin Hare (Waterloo, Wentang Kuo (Waterloo), Yu-Ru Liu (Waterloo), David McKinnon (Waterloo), Michael Rubinstein (Waterloo), Cam Stewart (Waterloo)

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Transcendence of various numbers and series
by
Michael Coons
Simon Fraser University
Coauthors: Peter Borwein

Abstract

The Liouville number, denoted l, is defined by

l:=0.100101001100011100001...,

where the nth bit is given by (1+l(n))/2; here l 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

g:=0.11011001110010011101100...

where the nth bit reflects the parity of the number of rational Gaussian primes dividing n, 1 for even and 0 for odd. We show that the Gaussian Liouville number and its relatives are transcendental. One such relative is the number (in series form)

∞
å
k=0

2^{3^k}

2^{3^k2}+2^{3^k}+1

The proofs involve showing the transcendence of formal power series arising as generating functions of completely multiplicative functions. This work is inspired by results of Dekking and Mahler.

Date received: May 27, 2008