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研究生:翁耀臨
研究生(外文):Yao-Lin Ong
論文名稱:庫麥爾同餘式及其推廣
論文名稱(外文):On Kummer''s Type Congruences and Generalizations
指導教授:余文卿余文卿引用關係
指導教授(外文):Minking Eie
學位類別:博士
校院名稱:國立中正大學
系所名稱:應用數學研究所
學門:數學及統計學門
學類:數學學類
論文種類:學術論文
論文出版年:2000
畢業學年度:88
語文別:英文
論文頁數:62
中文關鍵詞:庫麥爾同餘式Bernoulli 數Bernoulli 多項式Bernoulli 等式zeta 函數
外文關鍵詞:Kummer''s congruencesBernoulli numberBernoulli polynomialBernoulli identitieszeta functionGeneralized Bernoulli numberp-adic integration
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Eie在 [11] 發展了一套透過一些有理函數相對應的 zeta 函數產生 Bernoulli 等式的理論,提供了一系列有系統的方法來證明古典的庫麥爾同餘式以及一些庫麥爾型相關的同餘式或定理。
在這篇博士論文中,我們以上述的理論為基礎,產生了數個相關於特定三角級數和 Bernoulli 多項式之間的等式。另一方面,對於在 Bernoulli 數上的古典庫麥爾同餘式,我們提供了一個新的證明,並且把它推廣到在 Bernoulli 多項式上的庫麥爾同餘式。有趣的是,我們發現以同樣的論點幾乎可以對於所有文獻上的庫麥爾型同餘式重新給出一個簡捷的證明,並且可以做出類似於在 Bernoulli 多項式以及 Generalized Bernoulli 數上的庫麥爾型同餘式。最後我們以p-adic 積分的觀點重新詮釋庫麥爾型同餘式。
The theory of producing Bernoulli identities through zeta
functions associated with rational forms was initiated by Eie[11]. This provides a systematic method of proving the classical
Kummer''s congruences and some other theorems or congruences
related to Kummer''s type.
In this dissertation, we produced several identities among certain trigonometric series and Bernoulli polynomials based on this theory. On the other hand, we gave a new proof of Kummer''s
congruences on Bernoulli numbers and generalized the Kummer''s
congruences on Bernoulli numbers to Bernoulli polynomials.
Moreover, we found that it is interesting that almost all Kummer''s type congruences which were considered by L. Carlitz or H. S. Vandiver could be re-proved and generalized to the congruences on Bernoulli numbers or generalized Bernoulli numbers in the same way. Finally, another view of proving all Kummer''s type congruences via the $p$-adic integration was considered as well.
Cover
ABSTRACT
TABLE OF CONTENTS
INTRODUCTION
Chapter I. ZETA FUNCTIONS ASSOCIATED WITH RATIONAL FORMS
1.1. Basic Concept
1.2. Special Values of Zeta Functions
1.3. Several New Identities Among Sums of Certain Trigonometric Series and Bernoulli Polynomials
1.4. Two Ramanujan''s Identities
Chapter II. A NEW PROOF OF KUMMER''S CONGRUENCES
2.1. A New Proof of Xummer''s Congruences on Bernoulli Numbers
2.2. The Generalization of Kummer''s Congruences on Bernoulli Polynomials
2.3. The Exceptional Cases of Kummer''s Congruences
Chapter III. THEOREMS FOR VON STAUDT CLAUSSEN''S TYPE
3.1. A Generalization of von Staudt-Claussen''s Theorem
3.2. Congruent Relations of von Staudt-Claussen''s Type
Chapter IV. KUMMER''S TYPE CONGRUENCES
4.1. Kummer''s Type Congruences on Bernoulli Numbers
4.2. Kummer''s Type Congruences on Bernoulli Polynomials
4.3. Kummer''s Type Congruences on Generalized Bernoulli Numbers
Chapter V. p-ADIC INTEGRATION
5.1. p-adic Measures
5.2. p-adic L-function Lp(s;X)
REFERENCES
[1].T. M. Apostol ,Introduction to Analytic Number
Theory ,Springer-Verlag (1976)
[2].Bruce C. Berndt ,Ramanujan''s Notebooks Part I
and Part II ,Springer-Verlag (1985 and 1989)
[3].Z. I. Borevich and I. R. Shafarevich ,Number
Theory ,Academic Press (1966)
[4].L. Carlitz ,Some congruences for the Bernoulli numbers , Amer. J. Math, pp. 163-172 (1953)
[5] L.Carlitz, Some theorems on Kummer''s congruences ,vol 20 , Duke math. J , pp.423-432 (1953)
[6]L.Carlitz, Arithematic Properties of
Generalized Bernoulli Numbers, vol 202 ,J. Reine
Angew Math (1959)
[7]L.Carlitz, Criteria for Kummer''s Congruences, vol 6 ,Acta Arithmetica ,pp.375-391 (1961)
[8]L. Carlitz and H. Stevens,
Criteria for Generalized Kummer''s congruences, vol 207
J.Reine Angew Math, pp. 203-220 (1961)
[9]Minking Eie ,The special values at negative
integers of Dirichlet series associated with polynomials of
several variables, Proceedings of A.M.S. ,vol 119 ,pp. 51--61 (1993)
[10] Minking Eie, A Note on Bernoulli Numbers and
Shintani Generalized Bernoulli Polynomials ,Transactions of
A.M.S. ,pp. 1117-1136 ,vol 348 ,issue 3 (1996)
[11] Minking Eie and K. F. Lai, On Bernoulli
Identities and applications, Part I and II ,Revista
Metematica Iberoamericana ,vol 14 ,issue 1, pp. 167-213,
(1998)
[12]Minking Eie and Y. L. Ong , A
Generalization of Kummer''s Congruences ,vol 67 ,Abh. Math. Sem. Univ. Hamburg ,pp. 149-157 (1997)
[13]Minking Eie and Y. L. Ong, On Sums of Certain Trigonometric Series, manuscript (1996)
[14]Minking Eie and Y. L. Ong, Applications of Bernoulli Identities to Kummer''s Type Congruences, manuscript (1999)
[15]G. H. Hardy and E. M. Wright, An Introduction
to the Theory of Numbers ,Oxford University Press (1968)
[16]K. F. Irelend and M. I. Rosen ,A Classical
Introduction to Modern Number Theory ,Springer-Verlag ,
(1982)
[17]K. Iwasawa, Lectures on $p$-adic $L$-Functions
, Princeton University Press (1972)
[18]N. Jacobson, Basic Algebra I , W. H.
Freeman and Company (1985)
[19]N. Koblitz, $p$-adic Analysis: A Short Course
on Recent Work ,Cambridge University Press (1980)
[20]N. Koblitz, $p$-adic Numbers, $p$-adic Analysis and
Zeta-Functions ,Springer-Verlag (1977)
[21]E. E. Kummer ,Uber eine allegemeine
Eigenschaft der rational Entiwickelungscoefficienten einer
bestimmten Gattung analytischer Functionen ,J. Reine Angew
Math, pp. 368-372, vol 41, (1851)
[22]H. Rademacher ,Topics in Analytic NumberTheory, Springer-Verlag (1973)
[23]K. H. Rosen and W. M. Snyder ,
A Kummer Congruence for the Hurwitz-Herglotz Function
,vol 6 ,issue 1 ,Tokyo J. Math, pp. 125-133 (1983)
[24]J. J. Rotman, The Theory of Group: An
Introduction ,Allyn and Bacon, Ins. Boston (1965)
[25] J. P. Serre, A Course in Arithmetic,
Springer-Verlag (1973)
[26]H. R. Stevens, Bernoulli
numbers and Kummer''s Criterion , vol 24 , Fibonacci
Quart. , issue 2, pp.154-159 , (1986 )
[27]H. S. Vandiver, General
congruences involving the Bernoulli numbers, vol 28 ,Proc. Nat. Acad. Sci. U.S.A. pages 324-328 (1942)
[28]H. S. Vandiver, Certain
congruences involving the Bernoulli numbers, Duke math J.pp. 548-551 (1939)
[29]L. C. Washingtion ,Introduction to Cyclotomic
Fields, Springer-Verlag (1982)
[30]A. Weil, Number Theory (An approach through
history) ,Birkhauser Boston, Inc (1987)
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