Equidistribution of Gauss sums and Kloosterman sums

Let m be a positive integer. Fix a nontrivial additive character for each finite field F_q. To state the first result of this paper, we also fix r distinct multiplicative characters ₁,...,_r for each finite field F_q with more than r elements. We shall prove that when varies over multiplicative characters of F_q other than the m-th roots of the r-tuples of angles of Gauss sums are asymptotically equidistributed on the r-dimensional torus (S¹)^r as q goes to infinity.The n-dimensional Kloosterman sum over F_q at a F_q^× is One can define the angle (q,a) of Kl_n(q,a) in a suitable way. We shall prove that when a varies over nonzero elements of F_q, the q–1 angles (q,a^m) of Kloosterman sums are asymptotically equidistributed as q goes to infinity.Mathematics Subject Classification (2000) 11L05, 14F20