Fq[T]上多元算术函数的Ramanujan展开

Ramanujan Expansions of Arithmetic Functions of Several Variables over Fq[T]

从1976年到2017年,Wintner,Delange,Ushiroya和Toth逐步证明了定义在整数环上的多元算术函数都可以通过Ramanujan和加以展开.这类似于经典分析中周期函数的Fourier展开。本文主要研究了有限域上一元多项式环F_(q)[T]上Ramanujan和的性质,并证明了定义在F_(q)[T]上的多元算术函数也可以通过多项式Ramanujan和以及酉多项式Ramanujan和加以展开.

Combing Wintner,Delange,Ushiroya and Toth's works from 1976 to 2017,we have that the multi-variable arithmetic functions defined on integer ring can be expanded through the Ramanujan sums.This is an analogue of the Fourier expansion for periodic functions in the classical analysis.In this paper we further investigate the properties of Ramanujan sums in the polynomial ring F_(q)[T],and show that the multi-variable arithmetic functions defined on F_(q)[T]can also be expanded through the polynomial Ramanujan sums and the unitary polynomial Ramanujan sums.