On Estimation Of Some Trigonometrical Sums In Specific Interval

By the means of Vaughan identities in this paper, obtain to upper bound of two kinds of trigonometrical sums under certain conditions. The following two important conclusions are geted.Theorem 1 Let ? be a real number of the form ?=?/q+?/q2, with (?,q)=1,|?|?1,x,y are integer satisfy 3?y?x/(logx),r= log x, e(an) = e2?ian,we haveTheorem 2 Let (?)be a holomorphic cusp form of weight k for the full modular group SL2(Z) , (?),we have...