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

OEIS數列A065442

等價形式

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可以好容易證明到以下嘅和都係等於Erdős-Borwein 常數:

 
 
 
 

其中 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

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Erdős 喺 1948年證明咗呢個常數 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]

應用

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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

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  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.

外面連結

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