Mean Properties And Upper Bound Estimates Of Quasi Dedekind Sums

Dedekind sums is one of the important research objects of analytic number theory.Many experts and scholars at home and abroad have conducted extensive research on it.On this basis,hardy sums,Cochrane sums and so on,which are similar to Dedekind sums,are defined,which are collectively referred to as Dedekind-type sums.Generalized Dedekind sums,generalized hardy sums and generalized Cochrane sums are also defined,their arithmetic properties are studied,and many important conclusions are obtained.Based on generalized Dedekind sums and generalized Cochrane sums,this paper will define new Dedekind-type sums such as generalized Dedekind sums S(h,m,n,q,χ)and generalized Cochrane sums C(h,n,q,χ)with characteristics,and study their square mean.At the same time,the mixed mean of generalized Dedekind sums and Kloosterman sums and the upper bound estimation of generalized Cochrane sums with characteristics in general form are studied.Finally,Hardy sums is connected with the first kind of Chebyshev polynomials,the mean distribution of hardy sums on Chebyshev polynomials of the first kind is studied.Some asymptotic formulas are obtained by using the analysis method.The specific contents are as follows:1.Firstly,the definitions of generalized Dedekind sums with characteristics are given.By using the mean value theorem of Dirichlet L-function and the orthogonality of character sum,the square mean value is obtained.In addition,the mixed mean of generalized Dedekind sums and Kloosterman sums are discussed.Several interesting asymptotic formulas are obtained.2.The square mean of generalized Cochrane sum with characteristics is studied,and the following asymptotic formula is obtained:and according to the related properties of Dirichlet characteristics and the estimation of exponential sum,the upper bound estimation of generalized Cochrane sums with characteristics in general form is obtained:3.According to the first kind of Chebyshev polynomials and the related properties of Hardy sums,the asymptotic formula of the mean is obtained in the following form:...