And Research Issues Related To Bernoulli Polynomials

Bernoulli polynomial, which has many important properties, is a powerful tool for people to study other problems. Dedekind sum, which also has many important properties, plays a significant role in the study of modular function theory, and has been a hot topic in number theory research. The main purpose of this dissertation is to use the ralationship between Bernoulli polynomial and Dedekind sum, to generalize the classical Dedekind sum and its analogous sum which called Cochrane sum, and to study their arithmetic properties.Specific for, this paper mainly studies the following problems:1. First a generalized high-dimensional Cochrane sum Cn(h, k, q) is defined by using Bernoulli periodic function Bn(x), then its upper bound estimate is obtained through utilizing Weinstein’s estimate of the hyper-Kloosterman sum.2. The asymptotic property of the generalized high-dimensional Cochrane sum Cn(h,k,p) with p>3is studied, and an interesting mean square value formula is obtained.3. First a generalized homogeneous Dedekind sum Sn(a,b; q) is introduced by using Bernoulli periodic function Bn(x), then its distribution property in quarter intervals is studied though utilizing the estimate for character sums and the mean value theorem of Dirichlet L-functions, and a sharp asymptotic formula is obtained.