Character sums and the series L(1,χ) with applications to real quadratic fields

Abstract

In this article, let $k\equiv O$ or 1(mod4) be a fundamental discriminant, and let $\chi \left(n\right)$ be the real even primitive character modulo $k$. The series $L(1,\chi )={\displaystyle \sum }_{n=1}^{\infty }\frac{\chi \left(n\right)}{n}$ can be divided into groups of $k$ consecutive terms. Let $v$ be any nonnegative integer, $j$ an integer, $0\le j\le k-1$, and let $T(v,j,\chi )={\displaystyle \sum }_{n=j+1}^{j+k}\frac{\chi (vk+n)}{vk+n}$ Then $L(1,\chi )={{\displaystyle \sum }}_{v=0}^{\infty }T(v,0,\chi )={{\displaystyle \sum }}_{n=1}^j\chi \left(n\right)/n+{{\displaystyle \sum }}_{v=0}^{\infty }T(v,j,\chi )$. 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 for integer $v\ge \max \{1,\sqrt{k}/|M\left|\right\}$ if $M={{\displaystyle \sum }}_{m=1}^{j-1}\chi \left(m\right)+1/2\chi \left(j\right)\ne 0$ and $\left|M\right|\ge 3/2$. 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 $T(0,20,\chi )$.