| Let rκ(n)denote the number of representations of a natural num-ber n as the Sum of κ squares of integers;In[7]Fomenko considered the problem on the distribution of integral points on the cone Fomenko got the asymptotic formula where c=c(κ)>0 is a certain constant.When κ=4,Fomenko improved the resultIf f is a binary Hermitian form over O with associated positive definite quaternary quadratic form g over Z,where Ois the ring of integers of an imaginary quadratic number field K=Q((?))with discriminant D<0.[6]adopts the notation for the number of representations of the natural number n.Using the multiplicative property of number theory function, the analytic property of Dirichlet L function and Perron formula,the asymptotic formula of ∑n≤xσ(n)and∑n≤xσ2(n),and the asymptotic formula of∑n≤xσ(n)and ∑n≤xσ2(n)in the specific given restriction, this paper studies the mean value of the number of representations of natural number by positive definite quaternary quadratic forms.In this paper,we consider three special cases: number of representations of the natural number n by We denote the number of representations of the natural number n by number of representations of the natural number n by In this paper we establish the following results.Theorem 1.1(i)For x≥2,we have where A is a constant. where B is a constant.Theorem 1.2 (i) For x≥ 2, we have where C is a constant. where D is a constant.Theorem 1.3 (i) For x≥ 2, we have where E is a constant. where F is a constant. |
