Overview
The study focuses on deriving an asymptotic formula for a particular type of sum involving arithmetic functions. Specifically, it addresses sums where $A(n)$ represents either the $(1,n)$-th Fourier coefficients of an $SL(3,\mathbb{Z})$ Hecke-Maass cusp form, denoted $A(1,n)$, or the triple divisor function, denoted $d_3(n)$. The sums are structured as mixed powers of integers $n_1, n_2, n_3$ within specified ranges, combined with a binary quadratic polynomial $Q(x,y) \in \mathbb{Z}[x,y]$. The integer $k$ is constrained to be $k \geqslant3$. The function $\mathsf{a}(n)$ within the sum can be either the von-Mangoldt function or the identity function. In cases where $A(n) = A(1,n)$, the function $\mathsf{a}(n)$ can be any bounded arithmetical function.
Research Context
The core of this research involves the Fourier coefficients of $SL(3,\mathbb{Z})$ Hecke-Maass cusp forms, specifically the $(1,n)$-th coefficients, $A(1,n)$. Alternatively, the research considers the triple divisor function, $d_3(n)$. These functions are central to number theory and the study of automorphic forms. The asymptotic behavior of sums involving such coefficients is a key area of investigation, providing insights into their distribution and properties over arithmetic progressions or more complex structures.
Approach
The research established an asymptotic formula for sums of the form:
$ \mathop{\sum}_{\substack{1 \leqslant n_1, n_2 \leqslant X^{1/2} \\ 1 \leqslant n_3 \leqslant X^{1/k}}} A(Q(n_1,n_2) + n_3^k)\mathsf{a}(n_3) $
Here, $A(n)$ is defined as either the $(1,n)$-th Fourier coefficient of an $SL(3,\mathbb{Z})$ Hecke-Maass cusp form or the triple divisor function $d_3(n)$. The integer $k$ satisfies $k \geqslant3$. The binary quadratic polynomial $Q(x,y)$ is an element of $\mathbb{Z}[x,y]$. The summation ranges are given by $1 \leqslant n_1, n_2 \leqslant X^{1/2}$ and $1 \leqslant n_3 \leqslant X^{1/k}$. The function $\mathsf{a}(n)$ is specified as either the von-Mangoldt function or the identity function. A specific condition is noted for the case where $A(n)=A(1,n)$, under which $\mathsf{a}(n)$ may be any bounded arithmetical function.
Findings
The primary finding is the establishment of an asymptotic formula for the aforementioned sums. This formula provides a way to approximate the value of these sums for large $X$, indicating the leading order behavior and potentially lower-order terms. The conditions under which the formula holds are precisely defined by the nature of $A(n)$, the range of $k$, the form of $Q(x,y)$, and the choice of $\mathsf{a}(n)$.