Stoneham number

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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

$\alpha _{{b,c}}=\sum _{{n=c^{k}>1}}{\frac {1}{b^{n}n}}=\sum _{{k=1}}^{\infty }{\frac {1}{b^{{c^{k}}}c^{k}}}$

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.

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