Representations of integers by certain $2k$-ary quadratic forms

Abstract

Suppose $k$ is a positive integer. In this work, we establish formulas for for the number of representations of integers by the quadratic forms $$x_1^2+\cdots +x_k^2+l\left(x_{k+1}^2+\cdots +x_{2k}^2\right)$$ for $l\in \{2,4\}$.