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Abstract. Let p be an odd prime. Definewhere n is the multiplicative inverse of n modulo p such that 1 ≤ n ≤ p − 1. This paper shows that the sequence {e n } is a "good" pseudorandom sequence, by using the properties of exponential sums, character sums, Kloosterman sums and mean value theorems of Dirichlet L-functions.

For integers q, m, n, k with q, k ≥ 1, and Dirichlet character χ mod q, we define a mixed exponential sumwhere e(y) = e 2πiy , and a denotes the summation over all a with (a, q) = 1.The main purpose of this paper is to study the mean value of χ mod q q m=1 |C(m, n, k, χ; q)| 4 , and to give a related identity on the mean value of the general Kloosterman sum K(m, n, χ; q) := q a=1 χ(a)e ma + na q ,where aa ≡ 1 mod q.

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In this paper, we use the properties of Gauss sums, primitive characters and the mean value theorems of Dirichlet L-functions to study the hybrid mean value of Cochrane sums and general Kloosterman sums, and give two sharp asymptotic formulae.