Erdős-Borwein Constant
المؤلف:
Bailey, D. H. and Crandall, R. E.
المصدر:
"Random Generators and Normal Numbers." Exper. Math. 11
الجزء والصفحة:
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23-2-2020
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Erdős-Borwein Constant
The Erdős-Borwein constant 
, sometimes also denoted 
, is the sum of the reciprocals of the Mersenne numbers,
(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
 |
(8)
|
with 
is irrational.
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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