数论与特殊函数

Preface

Chapter 1 A quick introduction to complex analysis

1.1 Introduction

1.2 A quick introduction to complex analysis

1.2.1 Complex number system

1.2.2 Cauchy-Riemann equation and inverse functions

1.2.3 A rough description of complex analysis

1.2.4 Power series

1.2.5 Laurent expansion， residues

1.3 Around Jensen's formula

1.4 Partial fraction expansion

1.4.1 Partial fraction expansions for rational functions

1.4.2 Partial fraction expansion for the cotangent function and so its applications

Chapter 2 Elaboration of results of Srivastava and Choi

2.1 Glossary of symbols and formulas

2.2 Around the Hurwitz zeta-function

2.2.1 Applications of Proposition 2.1

2.2.2 Applications of Corollary 2.1

2.3 Euler integrals

2.4 Around the Euler integral

2.5 Around the Catalan constant

2.6 Kummer's Fourier series for the Log Gamma function

Chapter 3 Arithmetic Laurent coefficients

3.1 Introduction

3.2 Proof of results

3.3 Examples

3.4 The Piltz divisor problem

3.5 The partial integral Ik（x）

3.6 Generalized Euler constants and modular relation

Chapter 4 Mikolas results and their applications

4.1 From the Riemann zeta to the Hurwitz zeta

4.2 Introduction and the polylogarithm case

4.3 The derivative case

Chapter 5 Zeta-value relations

5.1 The structural elucidation of Eisenstein's formula

5.2 Proof of results

5.3 The Lipshitz-Lerch transcendent

Chapter 6 Summation formulas of Poisson and of Plana

6.1 The Poisson summation formula

6.2 Theta transformation formula and functional equation

6.3 The Hurwitz-Lerch zeta-function

6.4 Proof of results

Chapter 7 Modular relation and its applications

7.1 Introduction

7.2 The Riesz sum case

7.3 The Diophantine Dirichlet series

7.4 Elucidation of Katsurada's results

7.5 Proof of results

7.6 Modular relations

Bibliography

Index

李海龙的这本《数论与特殊函数（英文版）》分两部分，第一部分是复变函数理论和特殊函数的基础知识，可以作为基础部分或研究的原始资料（第一章内容）；第二部分是作者多年从事特殊函数的系统成果，这些成果发表在国内外著名杂志上15篇论文，全是SCI源期刊（第二章到第七章内容）。第二部分主要研究了Hurwitzzeta函数的部分和的积分渐进公式，推广了专著[SC]一些结果；研究了zeta函数和Riemannzeta函数洛朗系数的算术性质；研究了有理数域上一类zeta函数有关计算的伪问题，给出了Eisenstein公式的一般化，充分利用Lipcshitz-Lerchzeta函数的固有性质，简单和明了应用于特殊函数之中；给出了两个重要和公式体系，一个是对于C1函数的Poisson和，另一个是以用于Riemannzeta函数的功能方程的变换和公式，进一步推进Lipschitz-Lerchzeta函数的积分表示；得出了一些模关系结论，一个是对于周期贝努力多项式的傅里叶级数来历的研究，另一个是Katsurada结果的一般化的研究。