Some results on Waring-Goldbach type problems

Abstract

This thesis consists of three topics. The first one is on quadratic Waring-Goldbach problems. The second topic is about some additive problems involving fourth powers. The last topic is to consider an average result for the divisor problem in arithmetic progressions.

Chapter 1 is an introduction. In Chapter 2 the Lagrange equation with almost prime variables is studied, based on a method of Heath-Brown and Tolev. The techniques involve the circle method and the sieve method in analytic number theory. It is established that every sufficiently large integer, congruent to 4 modulo 24, can be represented as the sum of four squares of integers, each of which has at most four prime factors. In order to derive the desired result, it is important to study the asymptotic formula for a smooth counting function involving the sum of one square of a prime number and three squares of integers in arithmetic progressions. For this purpose, the square sieve of Heath-Brown and the Kloosterman refinement are employed. This is the main task of Chapter 3.

Chapter 4 is devoted to the investigation on the sum of four squares of primes and K powers of two. Inspired by the work of Wooley, this problem is dealt with by applying the linear sieve instead of the four dimensional vector sieve.

The theme of Chapter 5 is the three squares theorem with almost prime variables, firstly investigated by Blomer and Br?udern. A sharper result is obtained, in this chapter, by developing the three dimensional weighted sieve.

In Chapter 6, the main concern is the exceptional set for the sum of fourth powers. By developing Vaughan's p-adic iteration method, some new estimations are established. The new estimations improve upon the earlier result obtained by Kawada and Wooley. As a consequence of the new estimates, it is established that every sufficiently large integer under a natural congruence condition can be expressed as a sum of six fourth powers of primes and six fourth powers of integers. Moreover, a pair of diagonal quartic forms is considered. In Chapter 7, it is shown that a pair of diagonal quartic equations with 23 variables has nonzero integral solutions under certain reasonable conditions.

The purpose of Chapter 8 is to consider a Waring-Goldbach type problem involving twelve fourth powers of primes and one fourth power of a positive integer. The idea of proof is to combine Chen's switching principle and Iwaniec's linear sieve. A new result is obtained, which refines that of Ren and Tsang on this topic.

In the last chapter, a distribution problem for the divisor function on arithmetic progressions is investigated. Denote by ?q,b(X) the error term in the divisor problem in the arithmetic progression b(mod q). New asymptotic formulae for the variance A(q, X) =?qb=1|?q,b(x)|2 for X1=4+E ?· q?·X1=2-E and X1=2+E ? q ·?X1-E are derived. The distinct behaviors of A(q, X) in these two ranges are unveiled.