[1]唐志华,徐沥泉,徐利治.欧拉应用分析于数论研究综述[J].南京师大学报(自然科学版),2007,30(03):34-38.
　Tang Zhihua,Xu Liquan,HSU Leetsch Charles.Early Study in Which Leonhard Euler Made Use of the Mathematical Analysis to the Research of Number Theory[J].Journal of Nanjing Normal University(Natural Science Edition),2007,30(03):34-38.
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欧拉应用分析于数论研究综述()
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《南京师大学报（自然科学版）》[ISSN:1001-4616/CN:32-1239/N]

卷:
第30卷
期数:
2007年03期
页码:
34-38
栏目:
数学
出版日期:
2007-09-30

文章信息/Info

Title:
Early Study in Which Leonhard Euler Made Use of the Mathematical Analysis to the Research of Number Theory
作者:
唐志华1 ; 徐沥泉2 ; 徐利治3
(1. 江苏教育学院学前分院,江苏南京210004)
(2. 无锡市教育研究中心,江苏无锡214001)
(3. 大连理工大学数学系,辽宁大连116024)
Author(s):
Tang Zhihua1; Xu Liquan2; HSU Leetsch Charles3
1.Nanjing Higher Normal Preschool,Nanjing 210004,China
2. Wuxi Teaching and Research Center,Wuxi 214001, China)
3. Department ofMathematics, Dalian University of Technology, Dalan 116024, China
关键词:
L.欧拉;  数论;  伯努利数;  幂级数展开;  欧拉乘积
Keywords:
Leonhard Euler;  number theory; Bernoulli numbers;  power series expansion;  Eular p roduct
分类号:
O156
摘要:
扼要而又系统地综述了欧拉应用分析于数论研究的早期工作.其中有许多激动人心的数论公式与定理.例如,关于自然数方幂倒数的无穷和公式、关于Zeta函数的欧拉乘积公式、欧拉对4平方数定理的思考与证明,及其欧拉在解决这些问题的同时所创造的有关数论函数、分拆函数和理想数的概念等等.这些概念、定理或公式都是欧拉首先发现并加以精确论证的.与众不同的是,他善于把一个纯数论问题变换为一个分析问题,事实上欧拉的想法更具一般性.它足以展示欧拉的数学工作的深刻与广博.最后我们引述了欧拉发现的数论中几个著名的级数公式和二次互反性定律,它们都是欧拉在数论文库中留给我们的宝贵遗产.
Abstract:
Here p resented is a brief introduction of Euler’s study. There are many exciting formulas and theorems such as Euler’s infinite summation formula about recip rocal sum with powers of the natural num bers and Euler’s infinite p roduct rep resentation. Another examp le is Euler’s thinking and p roving about the p roof of Fermat’s four‘square theorem, yielding the arithmetical function, the partition function, and p rime ideal. These concep tions, theorems, and formulas were all first discovered accurately by Euler’s demonstrations. Eulerwas extraordinary at converting a p roblem of number theory into mathematical anal ysis. In fact, Euler’s ideas have become more generalized. These facts are enough to p rove that he had extensive and deep knowledge of his subject. Finally, we quoted a few famous examp les of power series, and the law of quadratic recip rocity which Euler found. They are all part of our p recious legacy in the li brary of number theory from Euler.

参考文献/References:

[ 1 ] 　ScharlauW, Hans Opolka. From Fermat to Minkowski [ C ] / / Lectures on Theory of Numbers and its Historical Development. New York: Sp ringer2Verlag, 1985: 13-31.
[ 2 ] 　KlineM. Mathematical Thought From Ancient toModern Times[M ]. New York: Oxford University Press, 1990.
[ 3 ] 　华罗庚, 王元. 数论在近似分析中的应用[M ]. 北京:科学出版社, 1978: 128-129.
[ 4 ] 　潘承洞,潘承彪. 初等代数数论[M ]. 济南:山东大学出版社, 1991: 112-117.
[ 5 ] 　徐利治. 论数学方法学[M ]. 济南: 山东教育出版社, 2001: 598-600.
[ 6 ] 　徐沥泉. 首屈一指的数学大师———纪念欧拉诞生300周年[ J ]. 自然杂志, 2007, 29 (3) : 1832186

备注/Memo

备注/Memo:
作者简介:唐志华(1965—) ,副教授,主要从事数学史、数学方法论和数学教育的研究. E-mail: tangzhihua@china. com
通讯联系人:徐利治(1920—) ,教授,中国科学院数学所顾问. 主要从事函数论和组合数学的研究.
更新日期/Last Update: 2013-05-05