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

E=\sum _{n=1}^{\infty }{\frac {1}{2^{n}-1}}\approx 1.606695152415291763\dots

（OEIS數列A065442）

等價形式

可以好容易證明到以下嘅和都係等於Erdős-Borwein 常數：

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}}}

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

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

其中 $\sigma _{0}(n)=d(n)$ 係除數函數，即係數 $n$ 有幾多個因數，係一個乘性函數嚟。要證明佢哋真係一樣，可以留意佢哋係寫咗做Lambert 級數嘅形式，所以可以調次序嚟加。^[1]

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

雖然佢係無理數，佢嘅二進制表達係可以好容易計到出嚟。^[4]^[5]

喺堆排序（heapsort）嘅最差情況分析入邊會出現Erdős-Borwein 常數，佢控制著將一個未排序陣列轉換成堆嘅時間。^[6]

↑ 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.
↑ Erdős, P. (1948), "On arithmetical properties of Lambert series" (PDF), J. Indian Math. Soc., New Series, 12: 63–66, MR 0029405.
↑ 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.
↑ 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.
↑ Crandall, Richard (2012), "The googol-th bit of the Erdős–Borwein constant", Integers, 12 (5): A23, doi:10.1515/integers-2012-0007, S2CID 122157888.
↑ Knuth, D. E. (1998), The Art of Computer Programming, Vol. 3: Sorting and Searching (第2版), Reading, MA: Addison-Wesley, pp. 153–155.