The Mean Value On Numbers Of Representation Of The Natural Number By Integers

Let r?(n)denote the number of representations of a natural num-ber n as the Sum of ? 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(?)>0 is a certain constant.When ?=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?x?(n)and?n?x?2(n),and the asymptotic formula of?n?x?(n)and ?n?x?2(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.