On the number of representations of integers by quaternary quadratic forms

Abstract

Let $R_1(n), R_2(n)$ denote the numbers of representations of a positive integer n by the quaternary quadratic forms $g_1(x_1,x_2,x_3,x_4)$ = $2( x_{1}^{2}+x_1 x_2+ x_{2}^{2})+2x_1x_3 +x_1x_4+ x_2x_3+2x_2x_4+2(x_{3}^{2}+x_3 x_4+x_{4}^{2}), g_2(x_{1},x_2,x_3,x_4)=8( x_{1}^{2}+x_{2}^{2})+x_{3}^{2}+x_{4}^{2}$, respectively, where $x_1$, $x_2$, $x_3$ and $x_4$ are integers. In this paper, we establish the asymptotic formulae for the sums $\sum \limits _{n\le x}R_i(n)$ and $\sum \limits _{n\le x}R_i^{2}(n)$ for $i=1,2$.