## Journal of the Mathematical Society of Japan

### Character sums and the series $L(1,\chi)$ with applications to real quadratic fields

Ming-Guang LEU

#### Abstract

In this article, let $k\equiv \mathrm{O}$ or 1(mod4) be a fundamental discriminant, and let $\chi(n)$ be the real even primitive character modulo $k$. The series $L(1,\displaystyle \chi)=\sum_{n=1}^{\infty}\frac{\chi(n)}{n}$ can be divided into groups of $k$ consecutive terms. Let $v$ be any nonnegative integer, $j$ an integer, $0\leq j\leq k-1$, and let $T(v,j,\displaystyle \chi)=\sum_{n=j+1}^{j+k}\frac{\chi(vk+n)}{vk+n}$ Then $L(1,\displaystyle \chi)=\sum_{v=0}^{\infty}T(v,0,\chi)=\sum_{n=1}^{j}\chi(n)/n+\sum_{v=0}^{\infty}T(v,j,\chi)$. In section 2, Theorems 2.1 and 2.2 reveal asurprising relation between incomplete character sums and partial sums of Dirichlet series. For example, we will prove that $T(v,j,\chi)\cdot M<\mathrm{O}$ for integer $v\displaystyle \geq\max\{1,\sqrt{k}/|M|\}$ if $M=\displaystyle \sum_{m=1}^{j-1}\chi(m)+1/2\chi(j)\neq 0$ and $|M|\geq 3/2$. In section 3, we will derive algorithm and formula for calculating the class number of a real quadratic field. In section 4, we will attempt to make a connection between two conjectures on real quadratic fields and the sign of $T(0,20,\chi)$.

#### Article information

Source
J. Math. Soc. Japan, Volume 51, Number 1 (1999), 151-166.

Dates
First available in Project Euclid: 10 June 2008

Permanent link to this document
https://projecteuclid.org/euclid.jmsj/1213108355

Digital Object Identifier
doi:10.2969/jmsj/05110151

Mathematical Reviews number (MathSciNet)
MR1661020

Zentralblatt MATH identifier
0940.11037

#### Citation

LEU, Ming-Guang. Character sums and the series $L(1,\chi)$ with applications to real quadratic fields. J. Math. Soc. Japan 51 (1999), no. 1, 151--166. doi:10.2969/jmsj/05110151. https://projecteuclid.org/euclid.jmsj/1213108355