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1. CMB Online first
Some normal numbers generated by arithmetic functions 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 ErdÅs 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 Categories:11K16, 11A63, 11N25, 11N37 |
2. CMB Online first
Some normal numbers generated by arithmetic functions 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 ErdÅs 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 Categories:11K16, 11A63, 11N25, 11N37 |
3. CMB 2007 (vol 50 pp. 399)
Expansions in Complex Bases Beginning with a seminal paper of R\'enyi, expansions in noninteger real bases have been widely
studied in the last
forty years. They turned out to be relevant in
various domains of mathematics, such as the theory of finite
automata, number
theory, fractals or dynamical systems.
Several results were extended by Dar\'oczy and K\'atai
for expansions
in complex bases. We introduce an adaptation of the so-called greedy
algorithm to the complex case, and we
generalize one of their main theorems.
Keywords:non-integer bases, greedy expansions, beta-expansions Categories:11A67, 11A63, 11B85 |
4. CMB 2002 (vol 45 pp. 115)
The Number of Non-Zero Digits of $n!$ Let $b$ be an integer with $b>1$. In this note, we prove that the
number of non-zero digits in the base $b$ representation of $n!$
grows at least as fast as a constant, depending on $b$, times $\log
n$.
Category:11A63 |
5. CMB 1999 (vol 42 pp. 68)
The Moments of the Sum-Of-Digits Function in Number Fields We consider the asymptotic behavior of the moments of the sum-of-digits
function of canonical number systems in number fields. Using Delange's
method we obtain the main term and smaller order terms which contain
periodic fluctuations.
Categories:11A63, 11N60 |