Stoneham number

In mathematics, the Stoneham numbers are a certain class of real numbers, named after mathematician Richard G. Stoneham (1920–1996). For coprime numbers b, c > 1, the Stoneham number α_b,c is defined as

$\alpha_{b,c} = \sum_{n=c^k>1} \frac{1}{b^nn} = \sum_{k=1}^\infty \frac{1}{b^{c^k}c^k}$

It was shown by Stoneham in 1973 that α_b,c is b-normal whenever c is an odd prime and b is a primitive root of c².

References

• Bugeaud, Yann (2012). Distribution modulo one and Diophantine approximation. Cambridge Tracts in Mathematics 193. Cambridge: Cambridge University Press. ISBN 978-0-521-11169-0. Zbl pre06066616.
• Stoneham, R.G. (1973). "On absolute $(j,ε)$-normality in the rational fractions with applications to normal numbers". Acta Arithmetica 22: 277–286. Zbl 0276.10028.
• Stoneham, R.G. (1973). "On the uniform ε-distribution of residues within the periods of rational fractions with applications to normal numbers". Acta Arithmetica 22: 371–389. Zbl 0276.10029.