NUMBER OF REPRESENTATIONS OF A POSITIVE INTEGER BY QUATERNARY AND OCTONARY QUADRATIC FORMS AND THE COEFFICIENTS RELATION

In this paper fora1,a2,···,ak∈N,k∈N, andn∈N∪{0}, we defineRk(a1,a2,···,ak;n) := card{(x1,x2,···,xk)∈Zk|n=a1x21+a2x22+···+akx2k}and then we obtainR4(1,1,3,3;n). Furthermore forn∈Nusing Legendre-Jacobi-Kroneckersymbol, we determine∑1≤m<n/3d1|md2|n−3m(−4d1)(−4d2)also we replace two integer coefficients appearing inR8(1,1,3,3,3,3,3,3;n),R8(1,1,1,1,3,3,3,3;n),andR8(1,1,1,1,1,1,3,3;n) by former simple coefficients.