On The Numbers Of Representation Of Integers By Quaternary Quadratic Forms

The theory of quadratic forms is one of central problems in number theory.In the literatures,many mathematicians such as Fermat,Lagrange,Gauss,Minkowski,Hasse and Siegel et al have made significant contributions to the representations of positive integers by quadratic forms.Let rk(n)denote the number of representations of a positive integer n by the quadratic form x12+x22+……+ xk2,i.e.n = x12+x22+…+xk2,(k? 2)where x1,x2,…,xk are integers.The average behavior of rk(n)is an old and interesting problem,which have also been studied by many authors(see Landau[16]).In the 1990's,Fomenko[6]and Muller[19,20]established,by the Rankin-Selberg convolution method,that for k>3,?rk2(n)= cxk-1+O(x(x(k-1)4k-3/4k-5),n<x where c = c(k)>0 is a constant.Fomenko[7]improved the estimate for k= 4,?r42(n)= 32?(3)x3 + O(x2(log x)5/3),n<x where?(s)is the Riemann zeta-function.Let f be a positive definite binary Hermitian form over ?,the ring ? of an imaginary quadratic number filed K = Q((>))with discrirminant D<0,where g is quaternary quadratic form associated with f.In[5],Elstrodt,Grunewald,and Mennicke adopt the notation R(n = f(c,d))= R(n = g(x1,x2,x3,x4))for the number of representations of the positive integer n.Wang and Lao[28]studied R(n)= R(n = 2(x12 +x1x2 + x22)+ x32+x3x4 + 3x42)and obtained?R2(n)= Ax3+O(x2(log x)11/3),n<x where A is a constant.In this paper we investigate another two special cases:(i)g1(x1,x2,x3,x4)?x12+x22+5(x23+x42).We denote the number of repre-sentations of the natural number n by R1(n)= R(n =x12 + x22+5(x31+x42)).(ii)g2(x1,x2,x3,x4)=x12+x22 + 8(x32 + x42).We denote the number of representations of the natural number n by R2(n)= R(n = x12+x22 + 8(x32 + x42)).In this paper we establish the following results.Theorem 1.For x>2,we have?R1(n)= C1x2 + O(x(logx)2/3),n?x where C1 is a constant.Theorem 2.For x ? 2,we have?R12(n)=C2x3 + O(x2(logx)3/11),n<x where C2 is a constant.Theorem 3.For x>2,we have?R2(n)= C3x2 + o(x logx),n<x where C3 is a constant.Theorem 4.For x>2,we have? R22(m)= C4X3 + O(x2(logax)5/3,n?x where C4 is a constant.