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The '''Erdős–Borwein [[Mathematical constant|constant]]''' is the sum of the [[Reciprocal (mathematics)|reciprocals]] of the [[Mersenne prime|Mersenne number]]s. It is named after [[Paul Erdős]] and [[Peter Borwein]].
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By definition it is:
 
:<math>E=\sum_{n=1}^{\infty}\frac{1}{2^n-1}</math>
which is approximately 1.60669 51524 15291 763… {{OEIS|A065442}}.
 
==Equivalent forms==
It can be proven that the following forms all sum to the same constant:
 
:<math>
E=\sum_{n=1}^{\infty}\frac{1}{2^{n^2}}\frac{2^n+1}{2^n-1}
</math>
 
:<math>
E=\sum_{m=1}^{\infty}\sum_{n=1}^{\infty} \frac{1}{2^{mn}}
</math>
 
:<math>
E=1+\sum_{n=1}^{\infty} \frac{1}{2^n(2^n-1)}
</math>
 
:<math>
E=\sum_{n=1}^{\infty}\frac{\sigma_0(n)}{2^n}
</math>
 
where σ<sub>0</sub>(''n'') = ''d''(''n'') is the [[divisor function]], a [[multiplicative function]] that equals the number of positive [[divisor]]s of the number ''n''. To prove the equivalence of these sums, note that they all take the form of [[Lambert series]] and can thus be resummed as such.<ref>The first of these forms is given by {{harvtxt|Knuth|1998}}, ex. 27, p. 157; Knuth attributes the transformation to this form to an 1828 work of [[Thomas Clausen (mathematician)|Clausen]].</ref>
 
==Irrationality==
Erdős in 1948 showed that the [[mathematical constant|constant]] ''E'' is an [[irrational number]].<ref>{{citation
| last = Erdős | first = P. | author-link = Paul Erdős
  | journal = J. Indian Math. Soc. (N.S.)
| mr = 0029405
| pages = 63–66
| title = On arithmetical properties of Lambert series
| url = http://www.renyi.hu/~p_erdos/1948-04.pdf
| volume = 12
| year = 1948}}.</ref> Later, Borwein provided an alternative proof.<ref>{{citation
| last = Borwein | first = Peter B. | authorlink = Peter Borwein
| doi = 10.1017/S030500410007081X
| issue = 1
| journal = Mathematical Proceedings of the Cambridge Philosophical Society
| mr = 1162938
| pages = 141–146
| title = On the irrationality of certain series
| volume = 112
| year = 1992}}.</ref>
 
Despite its irrationality, the [[Binary numeral system|binary representation]] of the Erdős–Borwein constant may be calculated efficiently.<ref>{{harvtxt|Knuth|1998}} observes that calculations of the constant may be performed using Clausen's series, which converges very rapidly, and credits this idea to [[John Wrench]].</ref><ref>{{citation
| last = Crandall | first = Richard | author-link = Richard Crandall
| doi = 10.1515/integers-2012-0007
| journal = Integers
| page = A23
| title = The googol-th bit of the Erdős–Borwein constant
| volume = 12
| year = 2012}}.</ref>
 
==Applications==
The Erdős–Borwein constant comes up in the [[average case analysis]] of the [[heapsort]] algorithm, where it controls the constant factor in the running time for converting an unsorted array of items into a heap.<ref>{{citation
| last = Knuth | first = D. E. | author-link = Donald Knuth
| edition = 2nd
| location = Reading, MA
| pages = 153–155
| publisher = Addison-Wesley
| title = [[The Art of Computer Programming]], Vol. 3: Sorting and Searching
| year = 1998}}.</ref>
 
==References==
{{reflist}}
 
==External links==
* {{MathWorld|urlname=Erdos-BorweinConstant|title=Erdos-Borwein Constant}}
 
{{DEFAULTSORT:Erdos-Borwein Constant}}
[[Category:Mathematical constants]]
[[Category:Irrational numbers]]
[[Category:Paul Erdős|Borwein constant]]

Latest revision as of 17:05, 5 December 2014

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