Journal of the Mathematical Society of Japan

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

Ming-Guang LEU

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In this article, let kO 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=1χ(n)n can be divided into groups of k consecutive terms. Let v be any nonnegative integer, j an integer, 0jk-1, and let T(v,j,χ)=n=j+1j+kχ(vk+n)vk+n Then L(1,χ)=v=0T(v,0,χ)=n=1jχ(n)/n+v=0T(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 vmax{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).

Article information

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

First available in Project Euclid: 10 June 2008

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 11M20: Real zeros of $L(s, \chi)$; results on $L(1, \chi)$ 11R11: Quadratic extensions 11R29: Class numbers, class groups, discriminants

Character sum Dirichlet series class number formula real quadratic field


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.

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