Distribution Of Primes In Sums Of K-th Powers And Related Problems

The sums of k-th powers x1k+x2k+…+xsk are hot topics in the study of analytic number theory,many scholars have a keen interest in it.Von Mangoldt function Λ(n)and Mobius function μ(n),which play an important role in the theory of prime distribution,many mathematicians work on problems related to Λ(n)and μ(n),and the corresponding asymptotic formulas and upper bounds are obtained.The problems related to the automorphic forms are popular problems in the current research,while the Fourier coefficients of automorphic forms are very important classes of arithmetic function,which are widely concerned by scholars.In this thesis,first of all,we study the sums of k-th powers of Λ(n),that is the distribution of prime numbers about the form m1k+m2k+…+msk,and prove the asymptotic formulas with s≥min(2k-1+1,1/2k(k+1)+1),we also establish the prime number theorem about m1k+m2k+…+msk,and under the suitable assumption a better minor term in the asymptotic formulas is obtained;secondly,we consider the sums of k-th powers related to μ(n)and the Fourier coefficients of the automorphic function,and get upper bounds for some suitable s.In Chapter 1,we introduce the background and main results of our research;In Chapter 2,we give the lemmas needed to prove the theorems;In Chapter 3,using the Circle Method and Bombieri-type theorem for exponential sums over primes,the asymptotic formulas related to Λ(n)are proved;In Chapter 4,the Circle Method is used to prove the upper bounds of the sums of k-th powers related to μ(n)and the Fourier coefficients of the automorphic function.