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Search: MSC category 11N37 ( Asymptotic results on arithmetic functions )

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1. 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 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
Categories: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 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
Categories:11K16, 11A63, 11N25, 11N37

3. CMB 2004 (vol 47 pp. 573)

Liu, Yu-Ru
A Generalization of the Turán Theorem and Its Applications
We axiomatize the main properties of the classical Turán Theorem in order to apply it to a general context. We provide applications in the cases of number fields, function fields, and geometrically irreducible varieties over a finite field.

Categories:11N37, 11N80

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