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
等價形式
編輯可以好容易證明到以下嘅和都係等於Erdős-Borwein 常數:
其中 係除數函數,即係數 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]
無理數
編輯應用
編輯喺堆排序(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]
參考資料
編輯- ↑ 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.