Mean value estimates in lattice point theory

Abstract

LetQ(u) be a positive definite quadratic form inr≥2 variables with a real symmetric coefficient matrix of determinantD. Given a real vectorb with 0≤b _j <1, forx>0 letA(x) be the number of lattice points in the ellipsoidQ(u+b)≤x, letV(x) be the volume of this ellipsoid andP(x)=A(x)−V(x). Let$M(x) = \int\limits_0^x {P^2 (y)dy}$. By introduction of a parameter ϖ we shall show how the treatment of estimates onP(x) and onM(x) can be unified.