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 some of 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 I_k(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
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