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Über dieses Buch

During the years 1903-1914, Ramanujan worked in almost complete isolation in India. During this time, he recorded most of his mathematical discoveries without proofs in notebooks. Although many of his results were already found in the literature, most were not. Almost a decade after Ramanujan's death in 1920, G.N. Watson and B.M. Wilson began to edit Ramanujan's notebooks, but they never completed the task. A photostat edition, with no editing, was published by the Tata Institute of Fundamental Research in Bombay in 1957. This book is the fourth of five volumes devoted to the editing of Ramanujan's notebooks. Parts I, II, and III, published in 1985, 1989, and 1991, contain accounts of Chapters 1-21 in Ramanujan's second notebook as well as a description of his quarterly reports. This is the first of two volumes devoted to proving the results found in the unorganized portions of the second notebook and in the third notebook. The author also proves those results in the first notebook that are not found in the second or third notebooks. For those results that are known, references in the literature are provided. Otherwise, complete proofs are given. Over 1/2 of the results in the notebooks are new. Many of them are so startling and different that there are no results akin to them in the literature.

Inhaltsverzeichnis

Frontmatter

Introduction

Abstract
Ramanujan built many castles. Although some may have been lost, most were preserved. Since his death in 1920, many mathematicians have been constructing the foundations for these magnificent structures. We continue this task in the present volume.
Bruce C. Berndt

Chapter 22. Elementary Results

Abstract
In this chapter we will examine Ramanujan’s findings that require only elementary high-school algebra for an understanding. A few of the results, e.g., Entries 5 and 32, however, require some knowledge of calculus for the proofs. Chapter 23, which examines Ramanujan’s discoveries in number theory, with the exception of his work on prime numbers, also contains some theorems that are accessible to those with only a background in elementary algebra. The latter results primarily involve equal sums of powers, and so it seems appropriate to place them in a chapter on number theory rather than in the present chapter. Possibly a few results in Chapter 22 fit better in Chapter 31 on elementary analysis, and possibly some would move a few results from that chapter to this chapter. Thus, the choice of material for this chapter is admittedly somewhat arbitrary.
Bruce C. Berndt

Chapter 23. Number Theory

Abstract
Ramanujan’s strong interest in number theory probably commenced no more than one or two years before his first letter to G. H. Hardy. Most of the results on number theory found in the notebooks lie in the 100 unorganized pages at the end of the second notebook and in the short third notebook consisting of 33 pages. Most of the material on these 133 pages was probably recorded in approximately the years 1912-1914, before Ramanujan departed for Cambridge. However, there is some evidence, especially from the material on number theory, that several entries in the third notebook were recorded in England.
Bruce C. Berndt

Chapter 24. Ramanujan’s Theory of Prime Numbers

Abstract
In his famous letters of 16 January 1913 and 29 February 1913 to G. H. Hardy, Ramanujan [23, pp. xxiii-xxx, 349–353] made several assertions about prime numbers, including formulas for π(x), the number of prime numbers less than or equal to x. Some of those formulas were analyzed by Hardy [3], [5, pp. 234–238] in 1937. A few years later, Hardy [7, Chapter II], in a very penetrating and lucid presentation, thoroughly discussed most of the results on primes found in these letters. In particular, Hardy related Ramanujan’s fascinating, but unsound, argument for deducing the prime number theorem. Generally, Ramanujan thought that his formulas for π(x) gave better approximations than they really did. As Hardy [7, p. 19] (Ramanujan [23, p. xxiv]) pointed out, some of Ramanujan’s faulty thinking arose from his assumption that all of the zeros of the Riemann zeta-function ζ(s) are real.
Bruce C. Berndt

Chapter 25. Theta-Functions and Modular Equations

Abstract
Chapters 16–21 in his second notebook [22] contain much of Ramanujan’s prodigious outpouring of discoveries about theta-functions and modular equations. However, the unorganized pages in the second and third notebooks also embrace a large amount of Ramanujan’s findings on these topics. In this chapter, we shall discuss most of this material. In Chapter 33 (Part V [9]), we relate Ramanujan’s fascinating theories of elliptic functions and modular equations with alternative bases. Chapter 26 contains Ramanujan’s theorems on inversion formulas for the lemniscate and allied integrals. Some material that normally would be placed in the present chapter is connected with continued fractions and so has been put in Chapter 32 (Part V [9]) on continued fractions.
Bruce C. Berndt

Chapter 26. Inversion Formulas for the Lemniscate and Allied Functions

Abstract
On pages 283, 285, and 286 in his second notebook [22], Ramanujan states ten inversion formulas for series. As an example, we quote (in a somewhat more compact notation) the entry at the top of page 285.
Bruce C. Berndt

Chapter 27. q-Series

Abstract
In view of Chapter 16 (Ramanujan [22], Berndt [6]), wherein several theorems on q-series are offered, and of the “lost notebook” (Ramanujan [24]), which is almost completely devoted to q-series, it is surprising that the unorganized portion of the second notebook and the third notebook contain only a small amount of material on q-series. In fact, only five identities are to be found here (Entries 1–5 below). More interesting are the asymptotic formulas given in Entries 6–8. Thus, in this chapter, only eight results are proved. Additional q-analysis can be found in Chapter 32, entitled Continued Fractions (Berndt [9]), or in the memoir of Andrews, Berndt, Jacobsen, and Lamphere [1]. Further related material can be found in Chapter 25 on theta-functions.
Bruce C. Berndt

Chapter 28. Integrals

Abstract
Integrals are scattered throughout Ramanujan’s notebooks. In particular, an abundance of integrals appears in Chapter 13 (Berndt [4]), which is highlighted by several elegant asymptotic expansions of integrals. In this chapter, we examine the nearly 50 results on integrals appearing in the 100 pages at the end of the second notebook, and in the 33-page third notebook.
Bruce C. Berndt

Chapter 29. Special Functions

Abstract
In this chapter, we collect together those results in the unorganized portions of the second and third notebooks that pertain to special functions. The first ten entries concern the gamma function. All ten entries are either known results or can easily be derived from standard theorems on the gamma function. The next four theorems arise from the theory of Bessel functions. These four theorems also are either classical or can be simply deduced from standard results on Bessel functions. The last section of the chapter is devoted to hypergeometric functions. By far, the most interesting result is contained in Section 15. Here Ramanujan offers a tantalizingly incomplete statement about a class of Saalschützian hypergeometric series. D. Bradley [1] has provided what is probably the best theorem that can be deduced from Ramanujan’s enigmatic statement. In particular, a large, new class of Saal-schützian series is summed in closed form. We complete this chapter by listing Ramanujan’s series for l/π, arising from certain hypergeometric functions.
Bruce C. Berndt

Chapter 30. Partial Fraction Expansions

Abstract
Ramanujan evidently had an affinity for partial fraction expansions, which can be found in several places in his notebooks. The heaviest concentrations lie in Chapters 14 and 18 and in the unorganized pages at the end of the second notebook. See our books [4] and [6] for accounts of the material in Chapters 14 and 18, respectively. In this chapter, we prove the 15 partial fraction decompositions found in the unorganized pages of the second notebook.
Bruce C. Berndt

Chapter 31. Elementary and Miscellaneous Analysis

Abstract
We have attempted to make the title of this chapter broad enough to encompass all the results contained therein. More precisely, we have placed in this chapter those results connected with analysis that are either very elementary or do not seem to fit in any other chapter. It must be admitted that some entries in this chapter are not particularly interesting. Some entries are trivial or are well known to almost all mathematicians, and one may ask why Ramanujan recorded them. There are at least two rejoinders. First, the third notebook appears to contain findings from Ramanujan’s youth, possibly as early as 1903, in addition to theorems perhaps recorded in England. Second, the recording of such items as the definition of a limit and the connection between a function’s singularities and the radius of convergence of its power series evince the absence of theoretical analysis in Ramanujan’s training.
Bruce C. Berndt

Backmatter

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