Stoneham number

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

${\displaystyle \alpha _{b,c}=\sum _{n=c^{k}>1}{\frac {1}{b^{n}n}}=\sum _{k=1}^{\infty }{\frac {1}{b^{c^{k}}c^{k}}}}${\displaystyle \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. In 2002, Bailey & Crandall showed that coprimality of b, c > 1 is sufficient for b-normality of α_b,c.^[2]

• Bailey, D. H.; Crandall, R. E. (2002), "Random generators and normal numbers" (PDF), Experimental Mathematics, 11 (4): 527–546, doi:10.1080/10586458.2002.10504704, S2CID 8944421 .
• Bugeaud, Yann (2012). Distribution modulo one and Diophantine approximation. Cambridge Tracts in Mathematics. Vol. 193. Cambridge: Cambridge University Press. ISBN 978-0-521-11169-0. Zbl 1260.11001.
• Stoneham, R.G. (1973). "On absolute $(j,ε)$-normality in the rational fractions with applications to normal numbers". Acta Arithmetica . 22 (3): 277–286. doi:10.4064/aa-22-3-277-286 . 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 (4): 371–389. doi:10.4064/aa-22-4-371-389 . Zbl 0276.10029.

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