The Erdős-Borwein constant , sometimes also denoted , is the sum of the reciprocals of the Mersenne
numbers,
(1)
(2)
(3)
(4)
(5)
(6)
(OEIS A065442), where is the number of divisors of and is a q-polygamma
function. The transformation from equation (1) to (2) follows from the series
transformation
(7)
due to Clausen in 1828 (Knuth 1998, pp. 155 and 157), with .
Erdős (1948) showed that the constant is irrational. Borwein
(1992) subsequently showed that
Bailey, D. H. and Crandall, R. E. "Random Generators and Normal Numbers." Exper. Math.11, 527-546, 2002.Borwein,
P. "On the Irrationality of Certain Series." Math. Proc. Cambridge Philos.
Soc.112, 141-146, 1992.Erdős, P. "On Arithmetical
Properties of Lambert Series." J. Indian Math. Soc.12, 63-66,
1948.Finch, S. R. Mathematical
Constants. Cambridge, England: Cambridge University Press, pp. 354-361,
2003.Knuth, D. E. The
Art of Computer Programming, Vol. 3: Sorting and Searching, 2nd ed.
Reading, MA: Addison-Wesley, 1998.Sloane, N. J. A. Sequence
A065442 in "The On-Line Encyclopedia
of Integer Sequences."