رامانوجان ۱۲۵: کنفرانس بین‌المللی به مناسبت ۱۲۵مین سالگرد تولد رامانوجان، ۵ تا ۷ نوامبر ۲۰۱۲، دانشگاه فلوریدا، گینزویل، فلوریدا

This volume contains the proceedings of an international conference to commemorate the 125th anniversary of Ramanujan's birth, held from November 5–7, 2012, at the University of Florida, Gainesville, Florida. Srinivasa Ramanujan was India's most famous mathematician. This volume contains research and survey papers describing recent and current developments in the areas of mathematics influenced by Ramanujan. The topics covered include modular forms, mock theta functions and harmonic Maass forms, continued fractions, partition inequalities, $q$-series, representations of affine Lie algebras and partition identities, highly composite numbers, analytic number theory and quadratic forms. Cover 1 Title page 4 Contents 0 Preface 8 Hecke grids and congruences for weakly holomorphic modular forms 12 1. Introduction 12 2. Preliminaries 15 3. Hecke grids on Γ*(1) 17 4. Hecke grids on Γ*(2) 20 5. Hecke grids on Γ*(3) 22 6. Hecke grids on Γ*(4) 24 References 26 Knots and q-series 28 1. Introduction 28 2. Background 29 3. Proof of (1.5) 31 4. Proof of (1.6) 31 5. Proof of (1.7) 32 6. The q-series for the ࠠ蔀 欀渀漀琀 33 7. Conclusion 35 References 35 A partition inequality involving products of two q-Pochhammer symbols 36 1. Introduction 36 2. Notation 39 3. The Injection 40 4. The Inverse 41 5. Examples 42 6. Proofs of the Dual and the Generalization 43 7. Two Invariants of the Injections 46 8. Conclusion 47 Acknowledgements 49 References 49 Analogues of Koshliakov’s formula 52 1. Introduction 52 2. Preliminary Results 54 3. Analogues of Koshliakov’s Formula 55 References 58 How to prove Ramanujan’s q-continued fractions 60 1. Introduction 60 2. Euler’s approach 61 3. The Rogers-Ramanujan Continued Fraction 61 4. Convergence matters 64 5. More continued fractions by Euler’s approach 68 6. The role of transformation formulas 71 7. A dose of insight into algebraical formulae 74 8. Infinite products as continued fractions 76 9. Conclusion 78 References 78 A nonsingular Z3 curve of genus 4 80 1. Introduction 80 2. Definition of the Surface 81 3. Inversion of Thomae’s Formulas and Other Identities 83 References 100 Ramanujan’s radial limits 102 1. Introduction and Statement of Results 102 Acknowledgements 105 2. Sketch of the proof of Theorem 1.2 105 3. Proof of Theorem 1.3 108 4. Discussion related to Theorem 1.3 109 References 112 An identity that may have changed the course of history 114 1. Introduction 114 2. Jacobi’s triple product identity 115 3. An important special case of Jacobi’s triple product identity 116 4. Obtaining Ramanujan’s identity 116 5. Additional comments 118 6. Notice 120 References 120 The major index generating function of standard Young tableaux of shapes of the form “staircase minus rectangle” 122 1. Introduction 122 2. The main result 124 3. A transformation formula for elliptic hypergeometric series and its consequences 127 References 132 Convergence of random continued fractions 134 1. Introduction and main result 134 2. Proof of Theorem 1.2 136 3. Some consequences 138 References 140 Tensor product decomposition of ̂sl(n) modules and identities 142 1. Introduction 142 2. Decomposition of V(Λ0)⊗V(Λi) 143 3. Generating Functions for Outer Multiplicities 147 4. Examples and Identities 152 References 154 Ramanujan, Robin, highly composite numbers, and the Riemann Hypothesis 156 1. Mathematical Background 156 2. Historical Survey 163 Acknowledgments 166 References 166 A quaternionic proof of the representation formulas of two quaternary quadratic forms 168 1. Introduction 168 2. Factorization of Lipschitz Type Quaternions 171 3. Analysis of 2-pure Primitive Factors 179 Acknowledgements 184 References 185 Back Cover 186 Krishnaswami Alladi, Frank Garvan, Ae Ja Yee, Editors. Conference In Honor Of Indian Mathematician Srinivasa Ramanujan Aiyangar. Includes Bibliographical References.