算子Lehmer问题与单位球面上的Mahler测度

This project will focus on the study of Lehmer’s problem in the number theory. The Lehmer problem is an open problem in the number theory with a nearly 100 years history which has deep connections with distinct mathematics. The applicant introduced the operator theory into the research on Lehmer’s problem, defined a new class of operators—subharmonic operators and defined the general operator Lehmer’s problem. These work makes Lehmer’s problem a problem in operator theory which is an original work. Based on operator theory research on Lehmer’s problem, this project try to focus on the operator Lehmer’s problem for some concrete operators on function spaces; Since the Mahler measure is the base of Lehmer’s problem, this project also aims to compute the Mahler measure on the unit sphere of some special polynomials with integer coefficients through operator theory and function theory. Related operator theory and function theory problems are also considered.

本项目将着重通过算子论来研究数论中的Lehmer问题。 Lehmer问题为数论中一个有近百年历史的公开问题，其与各种不同的数学分支都有深刻的联系。 申请人把算子论引入到数论中的著名公开问题的研究，定义了一类新的算子—次调和算子并且还定义了一般的算子Lehmer问题。找到了对这个公开问题的新的切入点和着力点，把这个问题变为一个算子论的问题。本项目立足于申请人对Lehmer问题的算子论研究，拟在已有的工作基础上研究具体的函数空间上的算子的算子Lehmer问题；Mahler测度是研究Lehmer问题的基础，所以本项目还将拟通过算子论和函数论中技巧来计算单位球面上的Mahler测度在某些特殊整系数多项式上的取值。也考虑与此相关的算子论和函数论问题。