Sum of the GL(3) Fourier coefficients over mixed powers.

Let 풜(n) be the (1,n)th Fourier coefficient of SL(3, ℤ) Hecke–Maass cusp form, denoted as A(1,n) or the triple divisor function, denoted as d3(n). Let k ≥ 3 be an integer. In this paper, we establish an asymptotic formula for the sum ∑ 1≤n1,n2≤X1/21≤n3≤X1/k풜(Q(n1,n2)+n3k)a(n 3), where a(n) is either von Mangoldt function or identity function, and Q(x,y) ∈ ℤ[x,y] is a binary quadratic polynomial. When 풜(n) = A(1,n), then a(n) can be any bounded arithmetical function. [ABSTRACT FROM AUTHOR]