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

Dedicated to Professor Takashi Ono on his seventieth birthday

抄録

In this article, let k≡ 0 or 1 (mod 4) be a fundamental discriminant, and let Χ(n) be the real even primitive character modulo k. The series \[ L(1, Χ)=∑_{n=1}^∞\frac{Χ(n)}{n} \] can be divided into groups of k consecutive terms. Let v be any nonnegative integer, j an integer, 0≤ j≤ k-1, and let \[T(v, j, Χ)=∑_{n=j+1}^j+k\frac{Χ(vk+n)}{vk+n} \] Then L(1, \displaystyle Χ)=∑_{v=0}^∞T(v, 0, Χ)=∑_{n=1}^jΧ(n)/n+∑_{v=0}^∞T(v, j, Χ).
In section 2, Theorems 2.1 and 2.2 reveal a surprising relation between incomplete character sums and partial sums of Dirichlet series. For example, we will prove that T(v, j, Χ)• M<0 for integer v\displaystyle ≥qmax{1, √{k}/|M|} if M=\displaystyle ∑_{m=1}^j-1Χ(m)+1/2Χ(j)≠ 0 and |M|≥q 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, Χ)$.