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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 Erdős 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 normal number, Euler function, sum-of-divisors function, Carmichael lambda-function, Champernowne's number
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
 

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