Abstract. The main purpose of this paper is using the elementary andanalytic methods to study the properties of the 2 k -th power mean valueof the generalized quadratic Gauss sums, and give two exact mean valueformulae for k = 3 and 4. 1. IntroductionLet q  2 be an integer, ˜ denotes a Dirichlet character modulo q . For anyinteger n , we de ne the generalized quadratic Gauss sums G ( n;˜ ; q ) as follows: G ( n;˜ ; q ) =∑ qa =1 ˜ ( a ) e ( na 2 q ) ; where e ( y ) = e 2 ˇiy . This sum is important, because it is a generalization ofthe classical quadratic Gauss sums G ( n;q ), which is de ned by G ( n ; q ) =∑ qa =1 e ( na 2 q ) : About the properties of G ( n;˜ ; q ), some authors had studied it, and obtainedmany interesting results. For example, for any integer n with ( n;q ) = 1, fromthe general result of Cochrane and Zheng [2] we can deduce that jG ( n;˜ ; q ) j  2 ! ( q ) q 12 ; where ! ( q ) denotes the number of all distinct prime divisors of q . The casewhere q is a prime is due to Weil [4]. Zhang [5] proved that for any odd prime