On the Gauss mean-value formula for class number

Abstract

In his masterwork Disquisitiones Arithmeticae, Gauss stated an approximate formula for the average of the class number for negative discriminants. 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 ∑n≤N H(-n) = 4π√N/(21ζ(3)) - 2π2 + O(N-15/44+∈) for every ∈ > 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 firstly the problem in a lattice point problem (as probably Gauss did) and we use a functional equation due to Shintani and Dirichlet class number formula to express the error term as a sum of character and exponential sums that can be estimated with techniques introduced in a previous work on the sphere problem.