On The Mean Of Dirichlet L-function And Original Feature Sum

Introduced by Dirichlet in studying the distribution of prime numbers,Dirichlet L-function is one of the important tools in the analytic number theory.The researches of Dirichlet L-function have a long history,and the results are numerous,for example,the zeros density theorem,the order estimation,and the mean value problems,which are still widely concerned by the majority of scholars.The studies of Dirichlet L-function not only have important value in theory and application,but also promote the researches on many topics in the analytic number theory,such as character sums,Dedekind sums.Based on the important conclusions of many scholars about L-function,in this thesis,we study the 2k-th mean value of L-function in the form of L(1,??)by using the properties of L-function,the relationship among L-function and the cotangent function and F(x),then obtain the corresponding asymptotic formulas.We further study the 2k-th mean value of primitive character sums over the incomplete interval[1,fq/p)by using the relationship between character sums and L-function.The main results obtained in this thesis are as follows:1.The 2k-th mean of L-function in the form of:(?)?(c)|L(1,??q)|2k,we obtain two asymptotic formulas by using different methods.2.The 2k-th mean of primitive character sums over[1,fq/p)in the form of:#12 when ?q(-1)=-1,we study 2k-th mean value of primitive character sums,when?q(-1)=1,we study the case where k=2.We obtain two asymptotic formulas by using the method of elementary number theory and analytic number theory.