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The main purpose of this paper is to use the properties of primitive characters, Gauss sums and Ramanujan’s sum to study the hybrid mean value of generalized Bernoulli numbers, general Kloosterman sums and Gauss sums, and give two asymptotic formulae.

HYBRID MEAN VALUE OF GENERALIZED BERNOULLI NUMBERS, GENERAL KLOOSTERMAN SUMS AND GAUSS SUMS