The Value Distribution Of Character Sums And Its Application

As is known to all,the Dirichlet character sums play a key role in the research of many number theory problems.For a long time,the research on character sums and the applications is one of the important research topics in number theory,especially in analytic number theory.The purpose of this thesis is to study the mean value and weighted mean value distribution of the character sums,and use the character sums to study the mean value distribution of the Dedekind type sums.More precisely,the main contents of this thesis can be summarized as follows:In the second chapter,firstly,we study the upper bound estimate of character sums on the set of the k-th power residues Ak.Secondly,we study the value dis-tribution of the character sums on the Ak from the perspective of the mean value.In other words,we use the mean value of Dirichlet L-functions and the related properties of k-th power residues and Gauss sums to study the estimate and mean value of the character sum as follows:(?)where x? 1 is an integer and p?5 is a prime.Futhermore,we study the weighted mean value of the Dirichlet L-functions and the general quadratic Gauss sums,and obtain some asymptotic formulae.It is of great significance for us to further under-stand the cancellation phenomenon and value distribution of the character sums on subsets.In the third chapter,we study the mean value of the Dedekind type sums with dual variables over short interval[1,N].It includes the general Dedekind sums and Hardy sums.Firstly,we connect the general Dedekind sums and Hardy sums with the Dirichlet character and L-functions,then use the mean value of L-functions and relat-ed properties of character sums to calculate the asymptotic formulae of the following sums:where l,k are two non-negative integers and b denote the multiplicative inverse of b modulo p.These results can help us to understand some interesting cancellations of Dedekind sums and Hardy sums.The fourth chapter is the application and promotion of character sums over in-complete intervals,we study the mean value of the Dedekind type sums.However,I.E.Shparlinski[75]mentioned that for long interval,for example,when N is equal to q/2,the corresponding asymptotic formula cannot be obtained by the method in his article.As supplement to the results of I.E.Shparlinski,this chapter study the hybrid mean value of Dirichlet L-functions with character sums over long interval,then combine the conversion methods,and the related properties of character sums to study the mean value of the general Dedekind sums and Hardy sums over inter-val[1,p/d).Finally,we obtain the asymptotic formulae of the following sums:where p>2 is a prime,d is a given prime with d<p.