Mean Value Problem Concerning Quadratic Forms And Distribution Of Primes In Special Arithmetic Progressions

In Chapter1of this thesis,we study the divisor number of the sequence m12+m22+m32.In[7],C.Calderon and M.J.devlasco proved In this paper,we will prove the followingTheorem1.1.Let S(x) be defined as in (0.0.2).We have where In Chapter2,we study distribution of primcs in the sequence m12+m22+m32. DefineTheorem2.1.For any fixed constant A>0,we have whereCollary. Let π2(x) denotes the prime numbers of the form m12+m22+m35not exceeding x,we haveIn Chapter3of this thesis,we study the mean-value estimates of the number of primes in arithmetic progressions to moduli with a power factor.Our result is the followingTheorem3.1.Let a be an integer, a≥2.If A>0,then there is a B=B(A)>0such that holds uniformly for moduli q=ak, q≤x2/5exp(-(loglogx)3).Theorem3.2. Let a be an integer, a≥2. If A>0, then there is a B=B(A)>0such that holds uniformly for moduli q=ak, q≤5/12exp(-(loglogx)3).This result improves the following result of Elliott [18] Let a be an integer, a>2. If A>0, then there is a B=B(A)>0such that holds uniformly for moduli q=ak, q≤1/3exp(-(loglogx)3).