On Exceptional Sets Of Squares Of Primes And A K-th Power Of A Prime

The Waring-Goldbach problem is a classic problem in number theory which attracts a lot of scholars. After Hardy and Littlewood introduced the circle method, research in this field developed rapidly. In 1938, Hua gave the upper bounds of a series of exceptional sets of the Waring-Goldbach problem. Later, Schwarz, Leung and Liu improved these upper bounds. In this paper, we improve some results of exceptional sets of integers, which are not restrict-ed by elementary congruence conditions and cannot be represented as sums of some prime powers either. We mainly concern about three kinds of exceptional sets of the Waring-Goldbach problem. For example, we prove that for k≥ 3, with at most O (N1-1/k2k-1+ε) exceptions, all positive integers n≤N, satisfying the necessary congruence conditions, are sums of two squares of primes and a k-th power of a prime; for k≥ 4, with at most O (N1/2-1/k2k-1+ε) exceptions, all positive integers n≤ N, satisfying the necessary congruence conditions, are sums of three squares of primes and a a k-th power of a prime and for k≥ 3, with at most O (N1/2-1/k2k-1+ε) exceptions, all positive integers n≤ N, satisfying the necessary congruence conditions, are sums of a prime, a square of a prime and a k-th power of a prime. These improve the previous results in this field.Briefly, let Aj (j= 3,4) and A be the sets of integers n satisfying the necessary congruence conditions, for all k≥ 2 and j= 3,4, we define Ej(N)= |εj(N)|and E(N)=|ε(N)|, where andε(N)={n ∈ A:n< N, n≠p1+p22+p3k for any prime pi, i= 1,2,3.}. Our results are as follows:Theorem 1. Let k ∈ N and E3(N) be defined as before. Then for all sufficient large N, we have when k≥ 3.Theorem 2. Let k ∈ N and E4(N) be defined as before. Then for all sufficient large N, we haveTheorem 3. Let k≥ 3 be an integer and E(N) be defined as before. Then for all sufficient large N, we have when k≥ 3.The previous best results of these problems came from Li [18], the corre-sponding results are as follows:We use the circle method to get our results and also use Wooley’s idea and Zhao’s lemma for exponential sums to deal with the minor arcs. This paper consists of five chapters. In the first chapter, we briefly introduce the research background and our main results. Then we state some lemmata we need in the second chapter. In Chapter 3, we show the outline of the circle method we used and give the proof of Theorem 2. Then we will prove Proposition 3.1, which is critical for us to prove Theorem 2. In the last chapter, we give the proofs of Theorem 1 and Theorem 3.