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The Googol-th Bit of the Erdős–Borwein Constant

Richard Crandall
Published Online: 2012-10-02 | DOI: https://doi.org/10.1515/integers-2012-0007


The Erdős–Borwein constant is a sum over all Mersenne reciprocals, namely

Although E is known to be irrational, the nature of the asymptotic distribution of its binary strings remains mysterious. Herein, we invoke nontrivial properties of the classical divisor function to establish that the googol-th bit, i.e., the bit in position to the right of the point is a `1.' Equivalently, the integer

is odd. Methods are also indicated for resolving contiguously the first bits (one full Terabyte) in about one day on modest machinery.

Keywords: Binary; Normality; Irrationality

About the article

Received: 2011-11-05

Accepted: 2012-02-25

Published Online: 2012-10-02

Published in Print: 2012-10-01

Citation Information: Integers, Volume 12, Issue 5, Pages 811–840, ISSN (Online) 1867-0652, ISSN (Print) 1867-0652, DOI: https://doi.org/10.1515/integers-2012-0007.

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© 2012 by Walter de Gruyter Berlin Boston.Get Permission

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