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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 1998 (vol 41 pp. 125)
Uniform approximation to Mahler's measure in several variables If $f(x_1,\dots,x_k)$ is a polynomial with complex coefficients, the Mahler measure
of $f$, $M(f)$ is defined to be the geometric mean of $|f|$ over the $k$-torus
$\Bbb T^k$. We construct a sequence of approximations $M_n(f)$ which satisfy
$-d2^{-n}\log 2 + \log M_n(f) \le \log M(f) \le \log M_n(f)$. We use these to prove
that $M(f)$ is a continuous function of the coefficients of $f$ for polynomials
of fixed total degree $d$. Since $M_n(f)$ can be computed in a finite number
of arithmetic operations from the coefficients of $f$ this also demonstrates
an effective (but impractical) method for computing $M(f)$ to arbitrary
accuracy.
Categories:11R06, 11K16, 11Y99 |