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

Abstract

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 $L(1, Χ)=∑_{n=1}<SUP>∞</SUP>\frac{Χ(n)}{n}$ can be divided into groups of k consecutive terms. Let v be any nonnegative integer, j an integer, 0≤ j≤ k-1, and let $T(v, j, Χ)=∑_{n=j+1}<SUP>j+k</SUP>\frac{Χ(vk+n)}{vk+n}$ Then L(1, \displaystyle Χ)=∑_{v=0}<SUP>∞</SUP>T(v, 0, Χ)=∑_{n=1}<SUP>j</SUP>Χ(n)/n+∑_{v=0}<SUP>∞</SUP>T(v, j, Χ).<br>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}<SUP>j-1</SUP>Χ(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)\$.

Journal

• Tokyo Sugaku Kaisya Zasshi

Tokyo Sugaku Kaisya Zasshi 51(1), 151-166, 1999-01

The Mathematical Society of Japan

References:  14

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Codes

• NII Article ID (NAID)
10002151462
• NII NACSIS-CAT ID (NCID)
AA0070177X
• Text Lang
ENG
• Article Type
ART
• ISSN
00255645
• NDL Article ID
4643300
• NDL Source Classification
ZM31(科学技術--数学)
• NDL Call No.
Z53-A209
• Data Source
CJP  NDL  J-STAGE

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