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The Googol-th Bit of the Erdős–Borwein Constant

1Center for Advanced Computation, Reed College, Portland, Oregon, USA

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

Publication History

Received:
2011-11-05
Accepted:
2012-02-25
Published Online:
2012-10-02

Abstract.

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

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