The class number formula for imaginary quadratic fields

Abstract

It is shown that the class number for negative discriminant $D$ can be expressed in terms of the base $B$ expansions of reduced fractions $\frac{x}{|D|}$, where $B$ is an integer prime to $D$. This result is then formulated to obtain information about the distribution of the values of $\chi (x)$, where $\chi$ is the quadratic character associated to $D$. This leads to simplified formulas for the class number in certain cases.