On The Generalized Cochrane And Identity, The Mean And Similar To Dedekind And Ramanujan And Some Identities,

In chapter 1, 2 and 3 we will study some properties of a sum analogous to general Dedekind sum. For any positive integer k and n and an arbitrary integer h, the classical Dedekind sum S(h, k) is defined bywithand the general Dedekind sum S(h,n, k) is defined bywhereBn(x) is the Bernoulli polynomial, Bn(x) defined on the interval 0 < x < 1 is the n-th Bernoulli periodic function. The various properties of S(h,n,k) were investigated by many authors. Professor Todd Cochrane introduced a sum analogous to the Dedekind sum,where a is defined by the equation aa = 1 mod k, denotes the summationover all a such that (a, fc) = 1. we define a generalized Cochrane sum as follows:and we define the Kloosterman sums K(m,n;q) as followed:where e(y) = e2iy. In this paper we will give a identity and two sharper asympotic formulas of the generalized Cochrane sum.In chapter 4 we will study the distribution properties of the hybrid mean value involving three sums analogous to Dedekind sums and Ramanujan sum.According Berndt, B.C.[10], three sums analogous to Dedekind sums were denned bywhere c > 0 is odd, d > 0 is even, andwhere c > 0 is even.and we define the Ramanujan's sum Rc(d) as followed:In this paper we will give four identities involving S(d,c), S1(d,c), Sz(d,c), S3(d,c) and Rc(d).