ON THE GAUSS MEAN-VALUE FORMULA FOR CLASS NUMBER

In his masterwork Disquisitiones Arithmeticae, Gauss stated an approxi mate formula for the average of the class number for negative discrimi nants. In this paper we improve the known estimates for the error term in Gauss approximate formula. Namely, our result can be written as N- 1 Sigma(n less than or equal to N) H(-n) = 4 pi root N/(21 zeta(3) - 2 pi(2) + O(N-15/44+epsilon) for every epsilon > 0, where H(-n) is, in modern notation, h(-4n). We also consider the average of h(-n) itself obtaining the same type of result. Proving this formula we transform f irstly the problem in a lattice point problem las probably Gauss did) and we use a functional equation due to Shintani and Dirichlet class n umber formula to express the error term as a sum of character and expo nential sums that can be estimated with techniques introduced in a pre vious work on the sphere problem.