Ramanujan's Lost Notebook: Part III

In the spring of 1976, George Andrews of Pennsylvania State University visited the library at Trinity College, Cambridge to examine the papers of the late G.N. Watson. Among these papers, Andrews discovered a sheaf of 138 pages in the handwriting of Srinivasa Ramanujan. This manuscript was soon designated, "Ramanujan's lost notebook." Its discovery has frequently been deemed the mathematical equivalent of finding Beethoven's tenth symphony. This volume is the third of five volumes that the authors plan to write on Ramanujan's lost notebook and other manuscripts and fragments found in The Lost Notebook and Other Unpublished Papers, published by Narosa in 1988. The ordinary partition function p(n) is the focus of this third volume. In particular, ranks, cranks, and congruences for p(n) are in the spotlight. Other topics include the Ramanujan tau-function, the Rogers–Ramanujan functions, highly composite numbers, and sums of powers of theta functions. Review from the second volume: "Fans of Ramanujan's mathematics are sure to be delighted by this book. While some of the content is taken directly from published papers, most chapters contain new material and some previously published proofs have been improved. Many entries are just begging for further study and will undoubtedly be inspiring research for decades to come. The next installment in this series is eagerly awaited." - MathSciNet Review from the first volume: "Andrews a nd Berndt are to be congratulated on the job they are doing. This is the first step...on the way to an understanding of the work of the genius Ramanujan. It should act as an inspiration to future generations of mathematicians to tackle a job that will never be complete." - Gazette of the Australian Mathematical Society This Volume Is The First Of Approximately Four Volumes Devoted To Providing Statements, Proofs, And Discussions Of All The Claims Made By Srinivasa Ramanujan In His Lost Notebook And All His Other Manuscripts And Letters Published With The Lost Notebook. In Addition To The Lost Notebook, This Publication Contains Copies Of Unpublished Manuscripts In The Oxford Library, In Particular, His Famous Unpublished Manuscript On The Partition And Tau-functions; Fragments Of Both Published And Unpublished Papers; Miscellaneous Sheets; And Ramanujan's Letters To G. H. Hardy, Written From Nursing Homes During Ramanujan's Final Two Years In England. This Volume Contains Accounts Of 442 Entries (counting Multiplicities) Made By Ramanujan In The Aforementioned Publication. The Present Authors Have Organized These Claims Into Eighteen Chapters, Containing Anywhere From Two Entries In Chapter 13 To Sixty-one Entries In Chapter 17. Most Of The Results Contained In Ramanujan's Lost Notebook Fall Under The Purview Of Q-series. These Include Mock Theta Functions, Theta Functions, Partial Theta Function Expansions, False Theta Functions, Identities Connected With The Rogers-fine Identity, Several Results In The Theory Of Partitions, Eisenstein Series, Modular Equations, The Rogers-ramanujan Continued Fraction, Other Q-continued Fractions, Asymptotic Expansions Of Q-series And Q-continued Fractions, Integrals Of Theta Functions, Integrals Of Q-products, And Incomplete Elliptic Integrals. Other Continued Fractions, Other Integrals, Infinite Series Identities, Dirichlet Series, Approximations, Arithmetic Functions, Numerical Calculations, Diophantine Equations, And Elementary Mathematics Are Some Of The Further Topics Examined By Ramanujan In His Lost Notebook. The Rogers-ramanujan Continued Fraction And Its Modular Properties -- Two-variable Generalizations Of (1.1.10) And (1.1.11) -- Hybrids Of (1.1.10) And (1.1.11) -- Factorizations Of (1.1.10) And (1.1.11) -- Modular Equations -- Theta-function Identities Of Degree 5 -- Refinements Of The Previous Identities -- Identities Involving The Parameter K = R(q)r[superscript 2](q[superscript 2]) -- Other Representations Of Theta Functions Involving R(q) -- Explicit Formulas Arising From (1.1.11) -- Explicit Evaluations Of The Rogers-ramanujan Continued Fraction -- Explicit Evaluations Using Eta-function Identities -- General Formulas For Evaluating R [characters Not Reproducible] And S [characters Not Reproducible] -- Page 210 Of Ramanujan's Lost Notebook -- Some Theta-function Identities -- Ramanujan's General Explicit Formulas For The Rogers-ramanujan Continued Fraction -- A Fragment On The Rogers-ramanujan And Cubic Continued Fractions -- The Rogers-ramanujan Continued Fraction -- The Theory Of Ramanujan's Cubic Continued Fraction -- Explicit Evaluations Of G(q) -- The Rogers-ramanujan Continued Fraction And Its Partitions And Lambert Series -- Connections With Partitions -- Further Identities Involving The Power Series Coefficients Of C(q) And 1/c(q) -- Generalized Lambert Series -- Further Q-series Representations For C(q) -- Finite Rogers-ramanujan Continued Fractions -- Finite Rogers-ramanujan Continued Fractions -- A Generalization Of Entry 5.2.1 -- Class Invariants -- A Finite Generalized Rogers-ramanujan Continued Fraction. Other Q-continued Fractions -- The Main Theorem -- A Second General Continued Fraction -- A Third General Continued Fraction -- A Transformation Formula -- Zeros -- Two Entries On Page 200 Of Ramanujan's Lost Notebook -- An Elementary Continued Fraction -- Asymptotic Formulas For Continued Fractions -- The Main Theorem -- Two Asymptotic Formulas Found On Page 45 Of Ramanujan's Lost Notebook -- An Asymptotic Formula For R(a,q) -- Ramanujan's Continued Fraction For (q[superscript 2];q[superscript 3])[infinity]/(q;q[superscript 3])[infinity] -- A Proof Of Ramanujan's Formula (8.1.2) -- The Special Case A = W Of (8.1.2) -- Two Continued Fractions Related To (q[superscript 2];q[superscript 3])[infinity]/(q;q[superscript 3])[infinity] -- An Asymptotic Expansion -- The Rogers-fine Identity -- Series Transformations -- The Series [characters Not Reproducible] -- The Series [characters Not Reproducible] -- The Series [characters Not Reproducible] -- An Empirical Study Of The Rogers-ramanujan Identities -- The First Argument -- The Second Argument -- The Third Argument -- The Fourth Argument -- Rogers-ramanujan-slater-type Identities -- Identities Associated With Modulus 5 -- Identities Associated With The Moduli 3, 6, And 12 -- Identities Associated With The Modulus 7 -- False Theta Functions -- Partial Fractions -- The Basic Partial Fractions -- Applications Of The Partial Fraction Decompositions -- Partial Fractions Plus -- Related Identities -- Remarks On The Partial Fraction Method -- Hadamard Products For Two Q-series. Stieltjes-wigert Polynomials -- The Hadamard Factorization -- Some Theta Series -- A Formal Power Series -- The Zeros Of K[subscript Infinity](zx) -- Small Zeros Of K[subscript Infinity](z) -- A New Polynomial Sequence -- The Zeros Of P[subscript N](a) -- A Theta Function Expansion -- Ramanujan's Product For P[subscript Infinity](a) -- Integrals Of Theta Functions -- Preliminary Results -- The Identities On Page 207 -- Integral Representations Of The Rogers-ramanujan Continued Fraction -- Incomplete Elliptic Integrals -- Preliminary Results -- Two Simpler Integrals -- Elliptic Integrals Of Order 5 (i) -- Elliptic Integrals Of Order 5 (ii) -- Elliptic Integrals Of Order 5 (iii) -- Elliptic Integrals Of Order 15 -- Elliptic Integrals Of Order 14 -- An Elliptic Integral Of Order 35 -- Constructions Of New Incomplete Elliptic Integral Identities -- Infinite Integrals Of Q-products -- Proofs -- Modular Equations In Ramanujan's Lost Notebook -- Eta-function Identities -- Summary Of Modular Equations Of Six Kinds -- A Fragment On Page 349 -- Fragments On Lambert Series -- Entries From The Two Fragments -- Location Guide -- Provenance. George E. Andrews, Bruce C. Berndt. Includes Bibliographical References And Indexes. In the spring of 1976, George Andrews of Pennsylvania State University visited the library at Trinity College, Cambridge, to examine the papers of the late G.N. Watson. Among these papers, Andrews discovered a sheaf of 138 pages in the handwriting of Srinivasa Ramanujan. This manuscript was soon designated, "Ramanujan's lost notebook." Its discovery has frequently been deemed the mathematical equivalent of finding Beethoven's tenth symphony. This volume is the fourth of five volumes that the authors plan to write on Ramanujan’s lost notebook. In contrast to the first three books on Ramanujan's Lost Notebook, the fourth book does not focus on q-series. Most of the entries examined in this volume fall under the purviews of number theory and classical analysis. Several incomplete manuscripts of Ramanujan published by Narosa with the lost notebook are discussed. Three of the partial manuscripts are on diophantine approximation, and others are in classical Fourier analysis and prime number theory. Most of the entries in number theory fall under the umbrella of classical analytic number theory. Perhaps the most intriguing entries are connected with the classical, unsolved circle and divisor problems. Review from the second volume: "Fans of Ramanujan's mathematics are sure to be delighted by this book. While some of the content is taken directly from published papers, most chapters contain new material and some previously published proofs have been improved. Many entries are just begging for further study and will undoubtedly be inspiring research for decades to come. The next installment in this series is eagerly awaited." - MathSciNet Review from the first volume: "Andrews and Berndt are to be congratulated on the job they are doing. This is the first step...on the way to an understanding of the work of the genius Ramanujan. It should act as an inspiration to future generations of mathematicians to tackle a job that will never be complete." - Gazette of the Australian Mathematical Society Annotation In the spring of 1976, George Andrews of Pennsylvania State University visited the library at Trinity College, Cambridge to examine the papers of the late G.N. Watson. Among these papers, Andrews discovered a sheaf of 138 pages in the handwriting of Srinivasa Ramanujan. This manuscript was soon designated, "Ramanujan's lost notebook." Its discovery has frequently been deemed the mathematical equivalent of finding Beethoven's tenth symphony. This volume is the third of five volumes that the authors plan to write on Ramanujans lost notebook and other manuscripts and fragments found in The Lost Notebook and Other Unpublished Papers, published by Narosa in 1988. The ordinary partition function p(n) is the focus of this third volume. In particular, ranks, cranks, and congruences for p(n) are in the spotlight. Other topics include the Ramanujan tau-function, the Rogers-Ramanujan functions, highly composite numbers, and sums of powers of theta functions. Review from the second volume: "Fans of Ramanujan's mathematics are sure to be delighted by this book. While some of the content is taken directly from published papers, most chapters contain new material and some previously published proofs have been improved. Many entries are just begging for further study and will undoubtedly be inspiring research for decades to come. The next installment in this series is eagerly awaited."--MathSciNet. Review from the first volume: "Andrews and Berndt are to be congratulated on the job they are doing. This is the first step ... on the way to an understanding of the work of the genius Ramanujan. It should act as an inspiration to future generations of mathematicians to tackle a job that will never be complete."--Gazette of the Australian Mathematical Society Front Matter....Pages I-XI Introduction....Pages 1-7 Ranks and Cranks, Part I....Pages 9-44 Ranks and Cranks, Part II....Pages 45-70 Ranks and Cranks, Part III....Pages 71-88 Ramanujan’s Unpublished Manuscript on the Partition and Tau Functions....Pages 89-180 Theorems about the Partition Function on Pages 189 and 182....Pages 181-204 Congruences for Generalized Tau Functions on Page 178....Pages 205-215 Ramanujan’s Forty Identities for the Rogers–Ramanujan Functions....Pages 217-335 Circular Summation....Pages 337-357 Highly Composite Numbers....Pages 359-402 Back Matter....Pages 403-435