# Erdős-Borwein 常數

Erdős-Borwein常數係所有梅森數嘅倒數和，係以數學家艾狄胥同Peter Borwein命名嘅。根據定義，可以計到

${\displaystyle E=\sum _{n=1}^{\infty }{\frac {1}{2^{n}-1}}\approx 1.606695152415291763\dots }$OEIS數列A065442

## 等價形式

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

## 無理數

Erdős 喺 1948年證明咗呢個常數${\displaystyle E}$係一個無理數[2]之後 Borwein 提供咗另一個證明。[3]

## 參考資料

1. 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.
2. Erdős, P. (1948), "On arithmetical properties of Lambert series" (PDF), J. Indian Math. Soc., New Series, 12: 63–66, MR 0029405.
3. Borwein, Peter B. (1992), "On the irrationality of certain series", Mathematical Proceedings of the Cambridge Philosophical Society, 112 (1): 141–146, doi:10.1017/S030500410007081X, MR 1162938.
4. 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.
5. Crandall, Richard (2012), "The googol-th bit of the Erdős–Borwein constant", Integers, 12 (5): A23, doi:10.1515/integers-2012-0007, S2CID 122157888.
6. Knuth, D. E. (1998), The Art of Computer Programming, Vol. 3: Sorting and Searching (第2版), Reading, MA: Addison-Wesley, pp. 153–155.

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