有关Ramanujan展开的结果的综述

A Survey on the Results of Ramanujan Expansion

摘要: Ramanujan和是现代数论中的一个重要工具，近年来也在信息科学中得到较多应用。这主要是基于匈牙利数论学家Wintner和1976年法国数论学家Delange的结果：整数环上的单变量算术函数都可以通过Ramanujan和加以展开。这类似于经典分析中的Fourier展开。随后这一结论被Ushiroya和匈牙利数论学家Tóth推广到了多变量情形。基于郑志勇教授的工作，最近我们证明了定义在有限域上一元多项式环 上的一大类算术函数(包括单变量和多变量情形)也可以通过Carlitz和Cohen定义的Ramanujan和加以展开。本文将对上面所得到的有关Ramanujan展开的结果进行综述。本文所有结果的证明都能在文末的参考文献中找到。

Abstract: The Ramanujan sum is an important tool in modern number theory, and it recently has been found many applications in information sciences. This is mainly because the results obtained by the Hungarian mathematician Wintner and the French mathematician Delange in 1976: the arithmetic functions in one variable defined on   can be expanded through the Ramanujan sums, which is similar with the Fourier expansions in the classical analysis. Subsequently, Ushiroya and Tóth generalized this result to the multi-variable cases. Based on the works of Zheng, recently we also proved that a large class of arithmetic functions in multi-variables defined on  , the pol-ynomial ring over a finite field, can also be expanded through the Ramanujan sums introduced by Carlitz and Cohen. This note will give a survey on the results of Ramanujan expansions mentioned above. All the proofs of the results in this note can be found in the references of this article.