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研究生:彭文紳
研究生(外文):Wen-Shen Peng
論文名稱:建立在VonSterneck函數上的推廣與應用
論文名稱(外文):On a generalization of the Von Sterneck function
指導教授:黃森山黃森山引用關係
指導教授(外文):Sen-Shan Huang
學位類別:碩士
校院名稱:國立彰化師範大學
系所名稱:數學系
學門:數學及統計學門
學類:數學學類
論文種類:學術論文
論文出版年:2002
畢業學年度:90
語文別:英文
論文頁數:29
中文關鍵詞:推廣的Von Sterneck函數推廣的拉瑪努江和推廣的Holder等式
外文關鍵詞:The generalized Von Sterneck functionThe generalized Ramamujan's sumThe generalized Holder identity
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在本篇論文,我們首先建立一個擴充的Von Sterneck 函數。經由漂亮的Holder's Identity啟發我們尋找一個更廣的對應關係,進而搜尋、確立這個問題答案的存在。最後,我們利用這推廣的式子建立了一些耐人尋味的美妙等式。

In this thesis, we start with generalizing the Von Sterneck function. According to Holder's identity, we establish a similar identity with respect to the generalized Von Sterneck function. Furthermore, we use the generalized Von Sterneck function to obtain some beautiful identities.

1. Introduction………………………………………….1
2. Preliminary……………………………………………4
3. The Main Result………………………………………9
4. Reference…………………………………………….28

﹝1﹞Douglas R. Anderson and T. M. Apostol, The evaluation of Ramamujan’s sum and generalizations, Duke Mathematical Journal, vol.20, 1953, 211-216.
﹝2﹞Eckford Cohen, An extension of Ramanujan’s sum, Duke Mathematical Journal, vol.16, 1949, 85-90.
﹝3﹞Eckford Cohen, An extension of Ramanujan’s sum II. Additive properties, Duke Mathematical Journal, vol.22, 1955 , 543-550.
﹝4﹞Eckford Cohen, An extension of Ramanujan’s sum III. Connections with totient functions. Duke Mathematical Journal, vol.23, 1956, 623-630.
﹝5﹞Eckford Cohen, Some totient functions, Duke Mathematical Journal, vol.23, 1956, 515-522.
﹝6﹞Eckford Cohen, An identity related to the Dedekind-Von Sterneck function, Duke Mathematical Journal, vol.29, 1962, 213-215.
﹝7﹞Dickson, L. E., History of the Theory of Numbers, vol.2, Chapter 3, Carnegie Inst. of Washington, Publication No.256, 1920.
﹝8﹞T. Estermann, On the representation of a number as the sum of three or more products, Proceedings of the Lomdon Mathematical Society ( 2) , vol. 34, 1932 , 190-195.
﹝9﹞G. H. Hardy, Note on Ramanujan’s trigonometrical function cq(n) , and certain series of arithmetical functions, Proceedings of the Cambridge Philosophical Society, vol. 20, 1921 , 263-271.
﹝10﹞G. H. Hardy, Ramanujan, Cambridge, 1940.
﹝11﹞O. Holder, Zur Theorie der Kreisteilungsgleichung Km(x)=0, Prace Mathematyczno Fizyczne, vol. 43(1936), 13-23.
﹝12﹞C. A. Nicol and H. S. Vandiver, A Von Sterneck arithmetical function and restricted partitions with respect to a modulus, Proceedings of the National Academy of Sciences, vol.40, 1954, 825-835.
﹝13﹞S. Ramanujan, On certain trigonometrical sums and their applications in the theory of numbers, Transactions of the Cambridge Philosophical Society, vol.22, no. 13, 1918 , 259-276.
﹝14﹞Sitzber. Akad. Wiss. Wien, Math.-Naturw. Klasse, 111(Abt. IIa), 1568-1601, 1902. Cf. also Bachmann, Niedere Zahlentheorie (Leipzig: B. G. Teubner, 1999), 2, 230-232.

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