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Erdős-Borwein Constant


The Erdős-Borwein constant E, sometimes also denoted alpha, is the sum of the reciprocals of the Mersenne numbers,

E=sum_(n=1)^(infty)1/(2^n-1)
(1)
=sum_(n=1)^(infty)1/(2^(n^2))(2^n+1)/(2^n-1)
(2)
=sum_(m=1)^(infty)sum_(n=1)^(infty)1/(2^(mn))
(3)
=sum_(n=1)^(infty)(sigma_0(n))/(2^n)
(4)
=1-(psi_(1/2)(1))/(ln2)
(5)
=1.606695152415291763...
(6)

(OEIS A065442), where sigma_0(n)=d(n) is the number of divisors of n and psi_q(z) is a q-polygamma function. The transformation from equation (1) to (2) follows from the series transformation

 sum_(n=1)^infty(x^n)/(1-x^n)=sum_(n=1)^infty(x^(n^2)(1+x^n))/(1-x^n)
(7)

due to Clausen in 1828 (Knuth 1998, pp. 155 and 157), with x=1/2.

Erdős (1948) showed that the constant E is irrational. Borwein (1992) subsequently showed that

 sum_(n=1)^infty1/(q^n-r)
(8)

with r!=0 is irrational.


See also

Erdős Number, Lambert Series, Mersenne Number, Tree Searching

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References

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."

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Erdős-Borwein Constant

Cite this as:

Weisstein, Eric W. "Erdős-Borwein Constant." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Erdos-BorweinConstant.html

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