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

Abstract

In this article, let k≡O or 1(mod4) be a fundamental discriminant, and let χ(n) be the real even primitive character modulo k. The series L(1,χ)=∑n=1∞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+1j+kχ(vk+n)vk+n Then L(1,χ)=∑v=0∞T(v,0,χ)=∑n=1jχ(n)/n+∑v=0∞T(v,j,χ). 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,χ)·M<O for integer v≥max{1,k/|M|} if M=∑m=1j-1χ(m)+1/2χ(j)≠0 and |M|≥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).

Locations

• Journal of the Mathematical Society of Japan - View - PDF