Research On The Mean Problem Of Generalized Dedekind And Hardy Sum

Dedekind sums, Hardy sums, Dirichlet L-functions, Bernouili polynomial play prominent role in the study of the analytic number theory and modular forms theory. So it is very meaningful to reveal the nature of Dedekind sums and Hardy sums. Many domestic and foreign scholars studied the properties of theirs. Thus on the basis of previous studies, we study two kind of hybrid mean value by using elementary number theory and analytic number theory related methods, and combining the estimates for character sums and mean value theorems of Dirichlet L-functions. The results are as follows:1. Let p> 2 be a prime, ? be any fixed integer, the positive integers N,l,k,n satisfies p^??N? p^1-?, l, k? 0, S?a,n,p? be the generalized Dedekind sums, we will study the following hybrid mean value and try to get a asymptotic formula.2. Let p> 2 be a prime, ? be any fixed integer, the positive integers N, l, k satisfies p^?? N? p^l-?, l,k? 0, H?a,p? be the Hardy sums, we will study the following hybrid mean value and try to get a asymptotic formula.