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

Let (n) be the (1 , n) th Fourier coefficient of S L (3 , ℤ) Hecke–Maass cusp form, denoted as A (1 , n) or the triple divisor function, denoted as d 3 (n). Let k ≥ 3 be an integer. In this paper, we establish an asymptotic formula for the sum ∑ 1 ≤ n 1 , n 2 ≤ X 1 / 2 1 ≤ n 3 ≤ X 1 / k (Q (n 1 , n 2) + n 3 k) 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 (n) = A (1 , n) , then a (n) can be any bounded arithmetical function. [ABSTRACT FROM AUTHOR]