关于四元非齐次幂和的除数问题

On the Sums of Divisor Problem with Quaternary Non-homogeneous Powers

设d(n)为Dirichlet除数函数,定义Sk(x)=∑1≤n2,n2,n3≤x^(1/2) 1≤n4≤x^(1/k)d(n2^(2)+n2^(2)+n3^(2)+n4^(k)),3≤k∈N.探讨了非齐次除数幂和Sk(x)的渐进性质,并建立了当k≥3时Sk(x)的渐近公式进一步深化了前人的结果.

Let d(n) denote the Dirichlet divisor function.Define Sk(x)=∑1≤n2,n2,n3≤x^(1/2) 1≤n4≤x^(1/k)d(n2^(2)+n2^(2)+n3^(2)+n4^(k)),3≤k∈N.In this paper,we discuss the asymptotic property of the sums of non-homogeneous powers Sk(x),and establish an asymptotic formula of Sk(x) for k≥3,which deepens the result of previous authors.