作　　者： ; ; ;

机构地区： 江苏教育学院

出　　处： 《南京师大学报（自然科学版）》 2007年第3期34-38,共5页

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. Here presented is a brief introduction of Euler＇ s study. There are many exciting formulas and theorems such as Euler＇ s infinite summation formula about reciprocal sum with powers of the natural numbers and Euler＇ s infinite product representation. Another example is Euler＇ s thinking and proving about the proof of Fermat＇ s four-square theorem, yielding the arithmetical function, the partition function, and prime ideal. These conceptions, theorems, and formulas were all first discovered accurately by Euler＇ s demonstrations. Euler was extraordinary at converting a problem of number theory into mathematical analysis. In fact, Euler＇ s ideas have become more generalized. These facts are enough to prove that he had extensive and deep knowledge of his subject. Finally, we quoted a few famous examples of power series, and the law of quadratic reciprocity which Euler found. They are all part of our precious legacy in the library of number theory from Euler.

关 键 词： 欧拉 数论 伯努利数 幂级数展开 欧拉乘积

领　　域： [理学] [理学]