On The Problem Concern The Integer Solutions Of General System Of Vinogradov’s Quadric Inside The Sphere

Vinogradov’s quadric is defined as the following pair of equations the more general system of this kind is defined as where (0.8) is the special case of (0.9), as the original form has been widely studied by a lot of researchers, and has also been generalized by other authors, they both obtained a lot of valuable results. (see paper [1], [2], [3]).Many authors considered the different forms of this problem and other related problems. The most original problem is the following equation: where k≥2 and k is a natural number, the problem is to get the integer solutions of (x1,x2,x3,x4) when satisfying|xi|≤N(i=1,2,3,4).When k take different values,the distribution of the integer solutions of equation (0.10) has opposite changes. When k≥3, Hooley(see paper [4],[5],[6]) has proved there are about 4N2 integer solutions in the diagonal lines,that is the integer solutions satisfying x1=x3,x2=x4 or x1=x4,x2=x3, the integer solutions that do not lying in the diagonal lines has lower order. But when k=2, the equation (0.10) has about N2 log N integer solutions in the rectangle, and the integer solutions lying in the diagonal lines is about N2, it has showed in this case the integer solutions lying in the diagonal lines does not dominate.Similarly, if we consider the integer solutions of (0.9) satisfying|xl,i|≤N. Let Vk(N) be the number of integer solutions satisfying (0.9). Then, according to (0.11), we can get then according to theorem(1.1) we have the following estimationBy using elementary method we could also get the asymptotic formula of Vk(N), but we would get weak results compared with analytic method. In fact, when k=2, we use Dirichlet’s hyperbolic method we could get following results: where c is a constant. Compare the results with (0.13), we could verify the conclusion remarked.In this paper we consider the distribution of the integer solutions of (0.9) when it satisfying some necessary conditions. Take Vk(N) as the number of integer solutions in the sphere satisfyingThrough more precise estimate of ζ(σ + it) in the horizotal line between 1/2≤σ≤1 or the mean-square estimate of ζ(σ+it), we have improved the latest research results by Valentin Blomer and Jorg Brudern(see the paper [7]), if we assume Lindelof hypothesis we could achieve better results .This paper is mainly composed of four parts.The first part introduces systematically the background of the subject and gives the result:Theorem 1.1 Let k≥ 2 an arbitrary fixed natural number.When k= 2,3,then for any real number δ satisfying we have Vk(N)=N3Pk(logN)+O(N3-δ)(0.12) where Pk is the polynomial of degree 2k-1 -1.Theorem 1.2 Let k ≥2 an arbitrary fixed natural number. When k≥ 4, then for any real number δ satisfying 0< δ <63/13·2k-1,we have Vk(N)=N3Pk(logN)+O(N3-δ)(0.13) where Pk is the polynomial of degree 2k-1-1.In particular, when k= 2,we have P2(x)= 48(x+c), where where χ is the nonprincipal character modulo 3,we also get better results: V2(N)=N3P2(logN)+O(N2 log N).(0.14)Theorem 1.3 Let k≥3 be any fixed natural number. If we assume the Lindelof hypothesis is true for ζ(s) and L(s,χ),then for any real number δ satisfying 0<δ<1,we have Vk(N) N3Pk(logN)+O(N3-δ).The second part introduces the preliminary knowledge needed to prove the theorem, including the mean-square estimate of ζ(σ+it), the truncated mellin integral formula, and Cauchy residue theorem to estimate the integral. According to the scope of k, to estimate the integral by using the estimate of ζ(σ+it) in the horizontal line or using Mean-square estimates ofζ(σ+it), choosing proper parameter, then we could achieve better results.The third part introduces the thoughts of Valentin Blomer and Jorg Brudern(see the paper [7]), we relate the calculation of integer solutions of (0.9) in the sphere with binary quadratic forms, and some application of sim-ple algebraic number theory, then transfer the theorem as the form of the lemma.The four part introduces the proof of these theorems. We mainly use the latest results of J. Bourgain in the estimate of ζ(1+it), coupled with Phragmen-Lindelof convexity principle and the mean-square estimate of it), then we get the main theorem in the paper.