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Some Normal Numbers Generated by Arithmetic Functions

Part of: Probabilistic theory: distribution modulo $1$; metric theory of algorithms Multiplicative number theory Elementary number theory

Published online by Cambridge University Press: 

Paul Pollack  and
Joseph Vandehey
Paul Pollack
Affiliation:
Department of Mathematics, Boyd Graduate Studies Research CenterUniversity of Georgia, Athens, GA 30602, USA Email: [email protected]@uga.edu
Joseph Vandehey
Affiliation:
Department of Mathematics, Boyd Graduate Studies Research CenterUniversity of Georgia, Athens, GA 30602, USA Email: [email protected]@uga.edu
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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 $0.235711131719\cdots \,$.

Keywords

normal numberEuler functionsum-of-divisors functionCarmichael lambda-functionChampernowne’s number

MSC classification

Primary: 11K16: Normal numbers, radix expansions, Pisot numbers, Salem numbers, good lattice points, etc.
Secondary: 11A63: Radix representation; digital problems 11N25: Distribution of integers with specified multiplicative constraints 11N37: Asymptotic results on arithmetic functions
Type
Article
Information
Canadian Mathematical Bulletin , First View , pp. 1 - 14
Copyright
© Canadian Mathematical Society 2014 

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