Abstract view
Paul Pollack,
Department of Mathematics, Boyd Graduate Studies Research Center, University of Georgia, Athens, GA 30602, USA
Joseph Vandehey,
Department of Mathematics, Boyd Graduate Studies Research Center, University of Georgia, Athens, GA 30602, USA
Abstract
Let $g \geq 2$. 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 $\phi$ denote Euler's totient function, let $\sigma$ be the sumofdivisors function, and let $\lambda$ be Carmichael's lambdafunction. We show that if $f$ is any function formed by composing $\phi$, $\sigma$, or $\lambda$, then the number
\[ 0. f(1) f(2) f(3) \dots \]
obtained by concatenating the base $g$ digits of successive $f$values is $g$normal. We also prove the same result if the inputs $1, 2, 3, \dots$ are replaced with the primes $2, 3, 5, \dots$. The proof is an adaptation of a method introduced by Copeland and ErdÅ‘s in 1946 to prove the $10$normality of $0.235711131719\ldots$.
MSC Classifications: 
11K16, 11A63, 11N25, 11N37 show english descriptions
Normal numbers, radix expansions, Pisot numbers, Salem numbers, good lattice points, etc. [See also 11A63] Radix representation; digital problems {For metric results, see 11K16} Distribution of integers with specified multiplicative constraints Asymptotic results on arithmetic functions
11K16  Normal numbers, radix expansions, Pisot numbers, Salem numbers, good lattice points, etc. [See also 11A63] 11A63  Radix representation; digital problems {For metric results, see 11K16} 11N25  Distribution of integers with specified multiplicative constraints 11N37  Asymptotic results on arithmetic functions
