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Erdős–Borwein constant

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Sum of the reciprocal of the Mersenne numbers

The Erdős–Borwein constant, named after Paul Erdős and Peter Borwein, is the sum of the reciprocals of the Mersenne numbers.

By definition it is:

E = ∑ n = 1 ∞ 1 2 n − 1 ≈ 1.606695152415291763 … {\displaystyle E=\sum _{n=1}^{\infty }{\frac {1}{2^{n}-1}}\approx 1.606695152415291763\dots }

Equivalent forms

It can be proven that the following forms all sum to the same constant:

E = ∑ n = 1 ∞ 1 2 n 2 2 n + 1 2 n − 1 {\displaystyle E=\sum _{n=1}^{\infty }{\frac {1}{2^{n^{2}}}}{\frac {2^{n}+1}{2^{n}-1}}}
E = ∑ m = 1 ∞ ∑ n = 1 ∞ 1 2 m n {\displaystyle E=\sum _{m=1}^{\infty }\sum _{n=1}^{\infty }{\frac {1}{2^{mn}}}}
E = 1 + ∑ n = 1 ∞ 1 2 n ( 2 n − 1 ) {\displaystyle E=1+\sum _{n=1}^{\infty }{\frac {1}{2^{n}(2^{n}-1)}}}
E = ∑ n = 1 ∞ σ 0 ( n ) 2 n {\displaystyle E=\sum _{n=1}^{\infty }{\frac {\sigma _{0}(n)}{2^{n}}}}

where σ0(n) = d(n) is the divisor function, a multiplicative function that equals the number of positive divisors 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.

Irrationality

In 1948, Erdős showed that the constant E is an irrational number. Later, Borwein provided an alternative proof.

Despite its irrationality, the binary representation of the Erdős–Borwein constant may be calculated efficiently.

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.

References

  1. (sequence A065442 in the OEIS)
  2. The first of these forms is given by Knuth (1998), ex. 27, p. 157; Knuth attributes the transformation to this form to an 1828 work of Clausen.
  3. Erdős, P. (1948), "On arithmetical properties of Lambert series" (PDF), J. Indian Math. Soc., New Series, 12: 63–66, MR 0029405.
  4. Borwein, Peter B. (1992), "On the irrationality of certain series", Mathematical Proceedings of the Cambridge Philosophical Society, 112 (1): 141–146, Bibcode:1992MPCPS.112..141B, doi:10.1017/S030500410007081X, MR 1162938, S2CID 123705311.
  5. 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.
  6. Crandall, Richard (2012), "The googol-th bit of the Erdős–Borwein constant", Integers, 12 (5): A23, doi:10.1515/integers-2012-0007, S2CID 122157888.
  7. Knuth, D. E. (1998), The Art of Computer Programming, Vol. 3: Sorting and Searching (2nd ed.), Reading, MA: Addison-Wesley, pp. 153–155.

External links

Irrational numbers
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