On class number formula for the real quadratic fields

Abstract

Let $k > 1$ be the fundamental discriminant, and let $\chi(n)$, $\varepsilon$ and $h$ be the real primitive character modulo $k$, the fundamental unit and the class number of the real quadratic field $\mathbf{Q}(\sqrt{k} )$, respectively. And let $[x]$ denote the greatest integer not greater than $x$.

In [3], M.-G. Leu showed $h = \big[ \sqrt{k} / (2\log{\varepsilon}) \sum_{n=1}^k \chi(n) / n \big] + 1$ for all $k$, and $h = \big[ \sqrt{k} / (2\log{\varepsilon}) \sum_{n=1}^{[k/2]} \chi(n) / n \big]$ in the case $k \neq m^2 + 4$ with $m \in \mathbf{Z}$.

In this paper we will show $h = \big[ \sqrt{k} / (2\log{\varepsilon}) \sum_{n=1}^{[k/2]} \chi(n) / n \big]$ for all fundamental discriminants $k > 1$.