[Number theory talk] On the average of class numbers of real quadratic fields

The average behaviour of class numbers of real quadratic fields is still poorly understood. Let $S(X)$ denote the sum of the class numbers of real quadratic fields with discriminant at most $X$. The class number formula gives the elementary bound $S(X)\ll X^{3/2}$, while a classical result of Siegel gives the best known upper bound $S(X)\ll X^{3/2}/\log X$. On the other hand, Hooley conjectured that $S(X)$ should be of order $X(\log X)^2$. There is therefore a substantial gap between what is known and what is expected. In this talk, I will discuss this problem and some recent progress on it.