Character sums and the series L(1,x) with applications to real quadratic fields
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- LEU Ming-Guang
- Department of Mathematics, National Central University
書誌事項
- タイトル別名
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- Character sums and the series L(1, Χ) with applications to real quadratic fields
- Dedicated to Professor Takashi Ono on his seventieth birthday
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抄録
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 \[ L(1, Χ)=∑_{n=1}∞\frac{Χ(n)}{n} \] can be divided into groups of k consecutive terms. Let v be any nonnegative integer, j an integer, 0≤ j≤ k-1, and let \[T(v, j, Χ)=∑_{n=j+1}j+k\frac{Χ(vk+n)}{vk+n} \] Then L(1, \displaystyle Χ)=∑_{v=0}∞T(v, 0, Χ)=∑_{n=1}jΧ(n)/n+∑_{v=0}∞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}j-1Χ(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)$.
収録刊行物
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- Journal of the Mathematical Society of Japan
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Journal of the Mathematical Society of Japan 51 (1), 151-166, 1999
一般社団法人 日本数学会
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詳細情報 詳細情報について
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- CRID
- 1390001205116075648
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- NII論文ID
- 10002151462
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- NII書誌ID
- AA0070177X
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- ISSN
- 18811167
- 18812333
- 00255645
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- MRID
- 1661020
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- NDL書誌ID
- 4643300
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- 本文言語コード
- en
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- データソース種別
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- JaLC
- NDL
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