Some New Results In Analytic Number Theory

In this dissertation,we consider several important problems in analytic number theory,such as on the seventh power moment of ?(x),on the fourth power moment of the error term for the divisor problem with congruence conditions,Waring-Goldbach problem with mixed powers,Waring-Goldbach problem with primes restricted in Piatetski-Shapiro sets.There are total four chapters in this dissertation and the contents of the chapters are as follows.In chapter one,we introduce the research background,the development history,and cur-rent results of the above problems.At the same time,we give the corresponding improvement conclusions.In chapter two,we study the seventh power moment of ?(x),and the fourth power mo-ment of the error term for the divisor problem with congruence conditions.For the first prob-lem,we improve Zhai's result,which was obtained in 2004,by using the technique to establish the estimate of the solutions of the Diophantine inequalities.For the second problem,we use stronger lower bounds of the linear combination of the quadratic irrational numbers.In chapter three,we study the Waring-Goldbach problem with mixed powers.We use the Hardy-Littlewood methods combining with the linear sieve methods of Iwaniec to improve the previous results.In chapter four,we study Waring-Goldbach problem with primes restricted in Piatetski-Shapiro sets.The key point of this problem is to estimate the exponential sums.By using Heath-Brown's identity,we can divide the exponential sums into type I sums and type II sums,and estimate them respectively.At last,we derive the desired results.