A generalization of Menon-Rao-Sury's identities to additive characters by Tóth's method

[摘要] In this paper, we obtain a new Menon-type identity: n ∑ a1,...,as ,b1,...,br=1 gcd(a1,...,as ,n)=1 gcd(a1 −c1,...,as −cs,b1,...,br,n) s λ1(b1)···λr(br) = ϕs(n)σr(gcd(w1,...,wr,n)), where λj(b) := exp(2πiwjb/n) is an additive character of Z/nZ for 1 ⩽ j ⩽ r, (c1,..., cs) ∈ Z s is a fixed vector such that gcd(c1,..., cs,n) = 1, ϕs(n) = n s ∏p|n (1 − p −s ) is the Jordan’s totient function and σr(n) = ∑d|n d r is the rth divisor function. This extends Rao’s identity ([9]) and Sury’s identity ([11]) to additive characters. Following the method of Toth [ ´ 14], we also generalize the above identity to arbitrary arithmetic function.

[学科分类] 数学（综合）

[关键词] Menon’s identity;arithmetic function;additive character;divisor function;Jordan’s totient function;Dirichlet convolution