On the average of class numbers of indefinite binary quadratic forms

We prove that the sum of class numbers of primitive indefinite binary quadratic forms with integer coefficients over positive nonsquare discriminants less than $X$ is $O(X^{3/2}\sqrt{\log\log X}/(\log X)^{3/2})$. This improves the longstanding classical upper bound $O(X^{3/2}/\log X)$ and implies th