Format:
Online-Ressource (XVII, 439 p. 1 illus, digital)
ISBN:
9781461440819
Series Statement:
SpringerLink
Content:
Preface -- 1 Introduction.- 2 Double Series of Bessel Functions and the Circle and Divisor Problems.- 3 Koshliakov's Formula and Guinand's Formula.- 4 Theorems Featuring the Gamma Function.- 5 Hypergeometric Series.- 6 Euler's Constant.- 7 Problems in Diophantine Approximation.- 8 Number Theory.- 9 Divisor Sums -- 10 Identities Related to the Riemann Zeta Function and Periodic Zeta Functions -- 11 Two Partial Unpublished Manuscripts on Sums Involving Primes.- 12 Infinite Series -- 13 A Partial Manuscript on Fourier and Laplace Transforms -- 14 Integral Analogues of Theta Functions adn Gauss Sums -- 15 Functional Equations for Products of Mellin Transforms -- 16 Infinite Products -- 17 A Preliminary Version of Ramanujan's Paper, On the Integral ∫_0^x tan^(-1)t)/t dt -- 18 A Partial Manuscript Connected with Ramanujan's Paper, Some Definite Integrals.- 19 Miscellaneous Results in Analysis -- 20 Elementary Results -- 21 A Strange, Enigmatic Partial Manuscript.- Location Guide -- Provenance -- References -- Index.
Content:
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.
Note:
Description based upon print version of record
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Preface; Contents; 1 Introduction; 2 Double Series of Bessel Functions and the Circle and Divisor Problems; 2.1 Introduction; 2.2 Proof of Ramanujan's First Bessel Function Identity (Original Form); 2.2.1 Identifying the Source of the Poles; 2.2.2 Large Values of n; 2.2.3 Small Values of n; 2.2.4 Further Reductions; 2.2.5 Refining the Range of Summation; 2.2.6 Short Exponential Sums; 2.2.7 Uniform Convergence When x Is Not an Integer; 2.2.8 The Case That x Is an Integer; 2.2.9 Estimating U2(a,b,T,); 2.2.10 Completion of the Proof of Entry 2.1.1
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2.3 Proof of Ramanujan's First Bessel FunctionIdentity (Symmetric Form)2.4 Proof of Ramanujan's Second Bessel Function Identity(with the Order of Summation Reversed); 2.4.1 Preliminary Results; 2.4.2 Reformulation of Entry 2.1.2; 2.4.3 The Convergence of (2.4.3); 2.4.4 Reformulation and Proof of Entry 2.1.2; 2.5 Proof of Ramanujan's Second Bessel Function Identity (Symmetric Form); 3 Koshliakov's Formula and Guinand's Formula; 3.1 Introduction; 3.2 Preliminary Results; 3.3 Guinand's Formula; 3.4 Kindred Formulas on Page 254 of the Lost Notebook; 4 Theorems Featuring the Gamma Function
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4.1 Introduction4.2 Three Integrals on Page 199; 4.3 Proofs of Entries 4.2.1 and 4.2.2; 4.4 Discussion of Entry 4.2.3; 4.5 An Asymptotic Expansion of the Gamma Function; 4.6 An Integral Arising in Stirling's Formula; 4.7 An Asymptotic Formula for h(x); 4.8 The Monotonicity of h(x); 4.9 Pages 214, 215; 5 Hypergeometric Series; 5.1 Introduction; 5.2 Background on Bilateral Series; 5.3 Proof of Entry 5.1.1; 5.4 Proof of Entry 5.1.2; 5.5 Background on Continued Fractions and Orthogonal Polynomials; 5.6 Background on the Hamburger Moment Problem; 5.7 The First Proof of Entry 5.1.5
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5.8 The Second Proof of Entry 5.1.55.9 Proof of Entry 5.1.2; 6 Two Partial Manuscripts on Euler's Constant ; 6.1 Introduction; 6.2 Theorems on and (s) in the First Manuscript; 6.3 Integral Representations of logx ; 6.4 A Formula for in the Second Manuscript; 6.5 Numerical Calculations; 7 Problems in Diophantine Approximation; 7.1 Introduction; 7.2 The First Manuscript; 7.2.1 An Unusual Diophantine Problem; 7.2.2 The Periodicity of vm; 7.3 A Manuscript on the Diophantine Approximation of e2/a; 7.3.1 Ramanujan's Claims; 7.3.2 Proofs of Ramanujan's Claims on Page 266
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7.4 The Third Manuscript8 Number Theory; 8.1 In Anticipation of Sathe and Selberg; 8.2 Dickman's Function; 8.3 A Formula for ( 0001975903/3003621EnFM1Fig1Print.tif12); 8.4 Sums of Powers; 8.5 Euler's Diophantine Equation a3+b3=c3+d3; 8.6 On the Divisors of N!; 8.7 Sums of Two Squares; 8.8 A Lattice Point Problem; 8.9 Mersenne Numbers; 9 Divisor Sums; 9.1 Introduction; 9.2 Ramanujan's Conclusion to trigsums; 9.3 Proofs and Commentary; 9.4 Two Further Pages on Divisors and Sums of Squares; 9.5 An Aborted Conclusion to trigsums?; 9.6 An Elementary Manuscript on the Divisor Function d(n)
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9.7 Thoughts on the Dirichlet Divisor Problem
Additional Edition:
ISBN 9781461440802
Additional Edition:
Druckausg. Andrews, George E., 1938 - Ramanujan's Lost Notebook ; 4 New York, NY : Springer, 2013 ISBN 9781461440802
Language:
English
Subjects:
Mathematics
DOI:
10.1007/978-1-4614-4081-9
URL:
Volltext
(lizenzpflichtig)
Author information:
Berndt, Bruce C. 1939-
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