Canadian Mathematical Society www.cms.math.ca
 location:  Publications → journals
Search results

Search: MSC category 11A63 ( Radix representation; digital problems {For metric results, see 11K16} )

 Expand all        Collapse all Results 1 - 5 of 5

1. CMB 2014 (vol 58 pp. 160)

Pollack, Paul; Vandehey, Joseph
 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 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 numberCategories:11K16, 11A63, 11N25, 11N37

2. CMB Online first

Pollack, Paul; Vandehey, Joseph
 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 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 numberCategories:11K16, 11A63, 11N25, 11N37

3. CMB 2007 (vol 50 pp. 399)

Komornik, Vilmos; Loreti, Paola
 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-expansionsCategories:11A67, 11A63, 11B85

4. CMB 2002 (vol 45 pp. 115)

Luca, Florian
 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)

Gittenberger, Bernhard; Thuswaldner, Jörg M.
 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
 top of page | contact us | privacy | site map |

© Canadian Mathematical Society, 2017 : https://cms.math.ca/