Constant Erdös - Borwein

Erdös – Borvein constant is a mathematical constant equal to the sum of the reciprocal of the Mersenne numbers . It is named after Pal Erdös and Peter Borwein , who established its key properties.

By definition, a constant is equal to:

E=\sum _{n=1}^{\infty }{\frac {1}{2^{n}-1}}

{\ displaystyle E = \ sum _ {n = 1} ^ {\ infty} {\ frac {1} {2 ^ {n} -1}}}

{\ displaystyle E = \ sum _ {n = 1} ^ {\ infty} {\ frac {1} {2 ^ {n} -1}}}

which is approximately 1, 606 695 152 415 291 763 783 301 523 190 924 580 480 579 671 505 756 435 778 079 553 691 418 420 743 486 690 565 711 801 670 155 576 ... ^[1] .

Content

Equivalent Forms

It can be shown that the following sums give the same constant:

E=\sum _{n=1}^{\infty }{\frac {1}{2^{n^{2}}}}{\frac {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=\sum _{m=1}^{\infty }\sum _{n=1}^{\infty }{\frac {1}{2^{mn}}}

{\ displaystyle E = \ sum _ {m = 1} ^ {\ infty} \ sum _ {n = 1} ^ {\ infty} {\ frac {1} {2 ^ {mn}}}}

,

E=1+\sum _{n=1}^{\infty }{\frac {1}{2^{n}(2^{n}-1)}}

{\ displaystyle E = 1 + \ sum _ {n = 1} ^ {\ infty} {\ frac {1} {2 ^ {n} (2 ^ {n} -1)}}}

,

E=\sum _{n=1}^{\infty }{\frac {\sigma _{0}(n)}{2^{n}}}

{\ displaystyle E = \ sum _ {n = 1} ^ {\ infty} {\ frac {\ sigma _ {0} (n)} {2 ^ {n}}}}

,

Where $\sigma _{0}(n)=d(n)$ ${\ displaystyle \ sigma _ {0} (n) = d (n)}$ Is the multiplicative divisor function equal to the number of positive divisors of the number $n$ ${\ displaystyle n}$ . To prove the equivalence of these formulas, we use the fact that they all represent the Lambert series ^[2] .

Irrationality

Erdös in 1948 showed that a constant is an irrational number ^[3] . Borvein later presented an alternative proof ^[4] .

Despite the irrationality, the binary representation of the constant is effectively calculated: Knut in the 1998 edition of The Art of Programming noted that the calculation can be done using the Clausen series, which converges very quickly ^[5] .

Applications

Erdös – Borvein constant arises in the analysis of the behavior of the pyramidal sorting algorithm ^[6]

Links

↑ sequence A065442 in OEIS
↑ The first of these formulas was introduced by Knut in 1998; In this case, Knut refers to the work of 1828 by Thomas Clausen
↑ Erdős, Pal (1948), " On arithmetical properties of Lambert series ", J. Indian Math. Soc. (NS) Vol . 12: 63–66 , < http://www.renyi.hu/~p_erdos/1948-04.pdf >
↑ Borwein, Peter B. (1992), " On the irrationality of certain series ", Mathematical Proceedings of the Cambridge Philosophical Society T. 112 (1): 141–146 , DOI 10.1017 / S030500410007081X
↑ Crandall, Richard (2012), " The googol-th bit of the Erdős – Borwein constant ", Integers T. 12: A23 , DOI 10.1515 / integers-2012-0007
↑ Knut, Donald (1998), The Art of Computer Programming , Vol. 3: Sorting and Searching (2nd ed.), Reading, MA: Addison-Wesley, p. 153–155 .

Literature

Weisstein, Eric W. Erdos-Borwein Constant on Wolfram MathWorld .

[1] sequence A065442 in OEIS

[2] The first of these formulas was introduced by Knut in 1998; In this case, Knut refers to the work of 1828 by Thomas Clausen

[3] Erdős, Pal (1948), " On arithmetical properties of Lambert series ", J. Indian Math. Soc. (NS) Vol . 12: 63–66 , < http://www.renyi.hu/~p_erdos/1948-04.pdf >

[4] Borwein, Peter B. (1992), " On the irrationality of certain series ", Mathematical Proceedings of the Cambridge Philosophical Society T. 112 (1): 141–146 , DOI 10.1017 / S030500410007081X

[5] Crandall, Richard (2012), " The googol-th bit of the Erdős – Borwein constant ", Integers T. 12: A23 , DOI 10.1515 / integers-2012-0007

[6] Knut, Donald (1998), The Art of Computer Programming , Vol. 3: Sorting and Searching (2nd ed.), Reading, MA: Addison-Wesley, p. 153–155 .