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

• LEU Ming-Guang

  Department of Mathematics, National Central University

書誌事項

タイトル別名

• Character sums and the series L(1, Χ) with applications to real quadratic fields
• Dedicated to Professor Takashi Ono on his seventieth birthday

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

• Ｊｏｕｒｎａｌ　ｏｆ　ｔｈｅ　Ｍａｔｈｅｍａｔｉｃａｌ　Ｓｏｃｉｅｔｙ　ｏｆ　Ｊａｐａｎ

  Ｊｏｕｒｎａｌ　ｏｆ　ｔｈｅ　Ｍａｔｈｅｍａｔｉｃａｌ　Ｓｏｃｉｅｔｙ　ｏｆ　Ｊａｐａｎ 51 (1), 151-166, 1999

  一般社団法人 日本数学会

キーワード

• Character sum
• Dirichlet series
• class number formula
• real quadratic field

詳細情報

• CRID: 1390001205116075648
• NII論文ID: 10002151462
• NII書誌ID: AA0070177X
• DOI: 10.2969/jmsj/05110151
• ISSN: 18811167
• ISSN: 18812333
• ISSN: 00255645
• MRID: 1661020
• NDL書誌ID: 4643300
• Web Site: https://ndlsearch.ndl.go.jp/books/R000000004-I4643300
• 本文言語コード: en
• データソース種別:
  • JaLC
  • NDL
  • Crossref
  • CiNii Articles
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