Mean value of some exponential sums and applications to Kloosterman sums

Let q, m, n, k be integers with q ⩾ 3 and k ⩾ 1 , define the exponential sum S ( m , n , k ; q ) = ∑ ′ a = 1 q e ( m a + n a k q ) , where e ( x ) = e 2 π i x , ∑ ′ a denotes the summation over all a such that ( a , q ) = 1 . In this paper, we shall study the fourth power mean ∑ ′ n = 1 q | S ( m , n , k ; q ) | 4 , and give some identities on the mean value of classical Kloosterman sums and hyper-Kloosterman sums.