Distribution of lattice points on surfaces of second order

Abstract

Let f1(x1,..., xl1) and f2(y1,...,yl2) be positive definite primitive quadratic forms in l1 and l2 variables, respectively. We obtain new results in the well-known problem on the number of lattice points on the cone f1(x1,...,xl1)=f2(y1,...,yl2), in the domain f1(x1,...,xl1)≦N for N»∞. Our technical tool is the Rankin-Selberg convolution. In several special cases the results can be sharpened by other methods. In addition, new facts concerning the uniform distribution of lattice points on ellipsoids in l variables, l odd, l≧5 are obtained.