Some Problems On Fourier Coefficients Of (?)-series

It is well known that the distribution of arithmetic function in arithmetic pro-gressions is one of the classical problems in analytic number theory,which is close-ly related to many problems,such as the Riemann hypothesis,the Goldbach con-jecture,the twin prime conjecture and so on.On the other hand,shifted convolu-tion sum is also an important topic and tool in modern number theory,which has many deep implications,for example,subconvexity and quantum unique ergodic-ity.It is an important topic in number theory to study Fourier coefficients of the(?)-series,which plays an important role in solving many classical problems,such as the Gauss circle problem,uniform distribution of lattice points and so on.In this thesis,we consider some problems involving Fourier coefficients of (?)-series.Namely,we will consider the Fourier coefficients r(n,Q)of (?)-series in arithmetic progressions and some shifted convolution sums involving the Fourier coefficients r(n,Q)of (?)-series.Firstly,we will study the Fourier coefficients r(n,Q)of (?)-series in arithmetic progressions.By means of functional equation and Mellin's transform,we shall es-tablish the Voronoi type summation formula for r(n,Q).After applying the Voronoi type summation formula for r(n,Q),analysing the complicated character sum and the integral,we obtain the asymptotic formula ofr(n,Q)in arithmetic progressions,which improves previous results.Further,using this asymptotic formula,we prove a large sieve type result for the Fourier coefficients r(n,Q)in arithmetic progres-sions.Secondly,we will consider some shifted convolution sums involving the Fouri-er coefficients r(n,Q)of (?)-series.Here three different approaches are devoted to generalize and improve previous results,and our results do not depend on the Ra-manujan conjecture.More precisely,·Using Jutila's circle method,the Voronoi summation formula of Fourier co-efficients of primitive cusp forms on GL(2)and the Voronoi type summation formula for r(n,Q),we obtain the upper bound of the GL(2)×GL(2)shifted convolution sums;·Using Kloosterman's circle method and the Voronoi summation formula of Fourier coefficients of GL(3)cusp forms,we derive the upper bound of the GL(3)×GL(2)shifted convolution sums;·Using the result of the Fourier coefficients of (?)-series in arithmetic progres-sion,Holder's inequality and Hua's inequality,we establish an asymptotic formula of the GL(4)× GL(2)shifted convolution sums.