A Study On The Mean Value Of Dedekind Sums And Kloosterman Sums

In analytic number theory,the mean value problems of Gauss sums,Dedekind sums,Ramanujan sums,Kloosterman sums and other famous sum-mations have always been an important research topic,and many scholars have made important achievement in this field.In this dissertation,we study the hybrid mean value of Dedekind sums and other summmations,the hy-brid mean value of Kloosterman sums and the quartic Gauss sums,and the mean value of the general 2-dimensional Kloosterman sums.Meanwhile,we also study some problems related to sequences and polynomials.The main content of this dissertation is organized as follows:1.The hybrid mean value of the Dedekind sums and some summations.In this dissertation,we propose an analogous Gauss sums and a general Ra-manujan sums.By using the properties of the classical Gauss sums and Dirichlet L-functions,we study the hybrid mean value of Dedekind sums and an analogous Gauss sums,Dedekind sums and a general Ramanujan sums,then obtain some related identities and asymptotic formulae.2.The mean value of the Kloosterman sums and their generalized sums.By using the analytical methods,we study the computational problems of the hybrid mean value of the Kloosterman sums and the quartic Gauss sums,and obtain a computational formula of the fourth hybrid power mean.Besides,we study the fourth power mean of the general 2-dimensional Kloosterman sums,then obtain an exact computational formula for it.3.Problems related to Smarandache-Pascal derived sequences and Chebyshev polynomials.By using combinatorial method and algebraic ma-nipulations,we study some problems related to the Smarandache-Pascal de-rived sequences and some computational problems of the power sum of the Chebyshev polynomials,then obtain some identities.