臺灣博碩士論文加值系統

利用狄克利雷特徵數求尤拉和在偶權數的取值

The classical~Euler sum is defined by
[
S_{p, q} := sum_{k=1}^{infty} frac{1}{k^{q}} sum_{j=1}^{k} frac{1}{j^{p}}.
]
In this thesis, we consider the more general sum defined by
[
S_{p, q}^{chi} := sum_{k=1}^{infty} frac{chi(k)}{k^{q}} sum_{j=1}^{k} frac{1}{j^{p}},
]
where $chi$ is~Dirichlet character. We have evaluations in terms of values at positive
integers of~Herwitz zeta functions, when $chi$ is odd and the weight $w = p+q$ is even.

á= ii
Abstract iii
1 Introduction and Notations 1
2 Some Basic Relations and Properties 6
3 The System of Equations F−B−R− can be Solvable and Some Results
9
4 Evaluations of F−1,2n+1,B−1,2n+1,R−1,2n+1, F−2n,2 and R−2,2n 30
Reference 34
i

[1] Bruce. C. Berndt, Ramanujan’s Notebook, Part I, Springer-Verlag, New
York, 1985.
[2] Minking Eie and Wen-Chin Liaw, Euler sums with Dirichlet Characters,
manuscript, July, 2005.
[3] Minking Eie and Yao Lin Ong, On the relations of extended Euler sums,
manuscript, February, 2004.
[4] Minking Eie, Wen-Chin Liaw and Fu-Yao Yang, On evaluations of generalized
Euler sums of even weight, Int. J. Number Theory.
[5] Dun-Ren Guo and Zhu-Xi Wang, Special Functions, World Scientific,
Singapore, 1989.
[6] Hans Rademacher, Topics in Analytic Number Theory, Springer-Verlag,
New York, 1980.
[7] Carl L. Siegel, Advanced Analytic Number Theory, Tata Institute of Fundamental
Research, Bombay, 1980.
[8] Lawrence C. Washington, Introduction to Cyclotomic Fields, second edition,
Springer-Verlag, New York, 1997.
[9] Fu Yao Yang, Bernoulli Identities and Application to Euler Sums,
Ph.D. Dissertation, National Chung Cheng University, 2004.