Proceedings of the Japan Academy, Series A, Mathematical Sciences

On class number formula for the real quadratic fields

Hiroki Sato

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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 $k > 1$.

Article information

Source
Proc. Japan Acad. Ser. A Math. Sci., Volume 80, Number 7 (2004), 129-130.

Dates
First available in Project Euclid: 18 May 2005

Permanent link to this document
https://projecteuclid.org/euclid.pja/1116442329

Digital Object Identifier
doi:10.3792/pjaa.80.129

Mathematical Reviews number (MathSciNet)
MR2094533

Zentralblatt MATH identifier
1068.11072

Subjects
Primary: 11R11: Quadratic extensions
Secondary: 11M06: $\zeta (s)$ and $L(s, \chi)$

Keywords
Class number real quadratic fields

Citation

Sato, Hiroki. On class number formula for the real quadratic fields. Proc. Japan Acad. Ser. A Math. Sci. 80 (2004), no. 7, 129--130. doi:10.3792/pjaa.80.129. https://projecteuclid.org/euclid.pja/1116442329


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References

  • Apostol, T. M.: Introduction to Analytic Number Theory. Undergraduate Texts in Mathematics. Springer-Verlag, New York-Berlin-Heidelberg-Tokyo (1976).
  • Cohn, H.: Advanced Number Theory. Dover Books on Advanced Mathematics. Dover, New York (1980).
  • Leu, M.-G.: On $L(1, \chi)$ and class number formula for the real quadratic fields. Proc. Japan Acad., 72A, 69–74 (1996).