The class number formula for imaginary quadratic fields

TL;DR

This paper presents a new formula expressing the class number of imaginary quadratic fields using base expansions of fractions, providing insights into the distribution of quadratic character values and simplifying calculations in specific cases.

Contribution

It introduces a novel expression for class numbers in terms of base expansions, linking number theory with digit expansion properties for the first time.

Findings

Derived simplified formulas for class numbers in specific cases

Connected base expansion properties with quadratic character distributions

Provided new tools for analyzing class numbers of 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.