Some Normal Numbers Generated by Arithmetic Functions

Pollack, Paul ; Vandehey, Joseph

Canadian mathematical bulletin, Tome 58 (2015) no. 1, pp. 160-173

Résumé

Let $g\,\ge \,2$ . A real number is said to be $g$ -normal if its base $g$ expansion contains every finite sequence of digits with the expected limiting frequency. Let $\varphi$ denote Euler’s totient function, let $\sigma$ be the sum-of-divisors function, and let $\lambda$ be Carmichael’s lambda-function. We show that if $f$ is any function formed by composing $\varphi$ , $\sigma$ , or $\lambda$ , then the number $$0.f\left( 1 \right)f\left( 2 \right)f\left( 3 \right)\,.\,.\,.$$ obtained by concatenating the base $g$ digits of successive $f$ -values is $g$ -normal. We also prove the same result if the inputs 1,2,3....are replaced with the primes 2, 3, 5.... The proof is an adaptation of a method introduced by Copeland and Erdõs in 1946 to prove the 10-normality of 0:235711131719...

Détail

DOI : 10.4153/CMB-2014-047-2

Mots-clés : 11K16, 11A63, 11N25, 11N37, normal number, Euler function, sum-of-divisors function, Carmichael lambda-function, Champernowne’s number