调和数的若干问题的研究

调和数是数论的重要研究对象，其研究有着悠久的历史，许多学者研究了调和数。.1991年，Eswarathasan和Levine猜想：对任给定的素数p，至多只有有限个n，使得第n个调和数的的分子能被p整除。本项目将研究: (a) Eswarathasan-Levine猜想；(b) 相邻的广义调和数分母的大小；(c) 第n个调和数的分母与1,2,...,n的最小公倍数的关系。申请人在此领域已有一定的工作积累，相关研究结果已发表在J. Number Theory，C. R. Acad. Sci. Paris, SerI等期刊上。特别地，申请人与导师陈永高教授证明了几乎所有相邻调和数的分母都相等。.本项目的研究将进一步揭示调和数的规律，对解析数论和组合数论的发展以及不同学科间的融合有着积极的作用。

Harmonic numbers are important research objects in number theory. The research of harmonic numbers has a long history and many scholars have studied harmonic numbers..In 1991, Eswarathasan and Levine conjectured that for any given prime number p, there are finitely many positive integers n such that the numerator of the n-th harmonic number can be divisible by p. The detailed study of the project are as follows: (a) the Eswarathasan-Levine conjecture; (b) the size of the denominators of adjacent generalized harmonic numbers; (c) the relationship between the denominator of the n-th harmonic number and the least common multiple of 1,2,...,n. The applicant has some accumulation in this field, the related research results have been published in J. Number Theory, C. R. Acad. Sci. Paris, Ser I, and so on. In particular, the applicant and her tutor professor Yong-Gao Chen have proved that the denominators of almost all adjacent harmonic numbers are equal. .The research of this project will further reveal the law of harmonic numbers, which also has a positive effect on the development of analytic number theory and combinatorial number theory, as well as the integration among different disciplines.