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January, 1999 Character sums and the series L(1,χ) with applications to real quadratic fields
Ming-Guang LEU
J. Math. Soc. Japan 51(1): 151-166 (January, 1999). DOI: 10.2969/jmsj/05110151


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).


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Ming-Guang LEU. "Character sums and the series L(1,χ) with applications to real quadratic fields." J. Math. Soc. Japan 51 (1) 151 - 166, January, 1999.


Published: January, 1999
First available in Project Euclid: 10 June 2008

zbMATH: 0940.11037
MathSciNet: MR1661020
Digital Object Identifier: 10.2969/jmsj/05110151

Primary: 11M20 , 11R11 , 11R29

Keywords: character sum , class number formula , Dirichlet series , real quadratic field

Rights: Copyright © 1999 Mathematical Society of Japan


Vol.51 • No. 1 • January, 1999
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