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

In this article, let k = 0 or 1 (mod4) be a fundamental discriminant, and let chi(n) be the real even primitive character module k. The series L(1,chi) = Sigma(n=1)(infinity) chi(n)/n can be divided into groups of k consecutive terms. Let v be any nonnegative integer, j an integer, 0 less than or equal to j less than or equal to k - 1, and let T(v, j, chi) = Sigma(n=j+1)(j+k) chi(vk + n)/vk + n Then L(1,chi) = Sigma(v=0)(infinity) T(V, 0, chi) = Sigma(n=1)(j) chi(n)/n + Sigma(v=0)(infinity) T(v, j, chi). 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, chi).M < 0 for integer v greater than or equal to max{1, root k//M/} if M = Sigma(m=1)(j-1) chi(m)+ 1/2 chi(j) not equal 0 and /M/ greater than or equal to 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).