Some Normal Numbers Generated by Arithmetic Functions

Abstract

Let $g\geqslant 2$. 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 $\unicode[STIX]{x1D711}$ denote Euler’s totient function, let $\unicode[STIX]{x1D70E}$ be the sum-of-divisors function, and let $\unicode[STIX]{x1D706}$ be Carmichael’s lambda-function. We show that if $f$ is any function formed by composing $\unicode[STIX]{x1D711}$, $\unicode[STIX]{x1D70E}$, or $\unicode[STIX]{x1D706}$, then the number

$$\begin{eqnarray}0.f(1)f(2)f(3)\cdots\end{eqnarray}$$

obtained by concatenating the base $g$ digits of successive $f$-values is $g$-normal. We also prove the same result if the inputs $1,2,3,\ldots$ are replaced with the primes $2,3,5,\ldots$. The proof is an adaptation of a method introduced by Copeland and Erdős in 1946 to prove the 10-normality of $0.235711131719\cdots \,$.