带有多个Dirichlet特征和加法特征的Menon-Sury恒等式

Menon-Sury’s Identity with Several Dirichlet Characters and Additive Characters

本文研究了同时带有多个Dirichlet特征和多个加法特征的Menon-Sury恒等式,给出了下列求和的明确表达式∑gcd(a1-1,…,as-1,b1,…,br,n)χ1(a1)…χs(as)λ1(b1)…λr(br),其中n是一个正整数,s,r为非负整数,Zn*是环Zn=Z/nZ的单位群,gcd(,)表示最大公因子,χi(1≤i≤s)是模n的导子为di的Dirichlet特征,λj(1≤j≤r)是Zn的加法特征.从有限交换群上的Fourier分析的角度看,我们的结果给出了这个算术函数f(a1,…,as,b1,…,br)=gcd(a1-1,…,as-1,b1,…,br,n)在交换群(Zn*)s×(Zn)r上的Fourier展开的系数的明确表达式.

This paper studies the Menon-Sury's identity with both Dirichlet characters and additive characters,and we shall give the explicit formula of the following sum∑gcd(a1-1,…,as-1,b1,…,br,n)χ1(a1)…χs(as)λ1(b1)…λr(br),where n is a positive integer,s,r are nonnegative integers,Zn*is the group of units of the ring Zn=Z/nZ,gcd(,)represents the greatest common divisor,χi(1≤i≤s)are Dirichlet characters mod n with conductors di,λj(1≤j≤r)are additive characters of Zn.From the point of view of Fourier analysis on finite Abelian groups,our result presents the explicit expression of Fourier coefficients of the function f(a1,…,as,b1,…,br)=gcd(a1-1,…,as-1,b1,…,br,n)on the Abelian group(Zn*)s×(Zn)r.