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# Some normal numbers generated by arithmetic functions

• 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
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## 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 sum-of-divisors function, and let $\lambda$ be Carmichael's lambda-function. 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 $0.235711131719\ldots$.
 Keywords: normal number, Euler function, sum-of-divisors function, Carmichael lambda-function, Champernowne's number
 MSC Classifications: 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

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