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Number theory in the spirit of Ramanujan. (English) Zbl 1117.11001

Student Mathematical Library 34. Providence, RI: American Mathematical Society (AMS) (ISBN 0-8218-4178-5/pbk). xix, 187 p. (2006).
Srinivasa Ramanujan was widely understood to be an important mathematician at least by 1918, the year of his election as a Fellow of the Royal Society. Recognition of the significance of his work is far greater today than it was at the time of his death 87 years ago, at the age of 32. Thus the recent publication of “Number theory in the spirit of Ramanujan” comes as no surprise.
Any mathematican with as much as a passing interest in the work of Ramanujan is certain to be familiar with the name of the author. By virtue of his prodigious contribution to our understanding of Ramanujan’s legacy, the author has earned a place in mathematical history. As if his five-volume series on Ramanujan’s notebooks (Springer Verlag, 1985–1998, see Part I (1985; Zbl 0555.10001), Part II (1989; Zbl 0716.11001), Part III (1991; Zbl 0733.11001), Part IV (1994; Zbl 0785.11001) and Part V (1998; Zbl 0886.11001) were not sufficient contribution to mathematics, he has undertaken, jointly with the estimable George Andrews, a multiple-volume series on the Lost Notebooks [Vol. I, New York: Springer (2005; Zbl 1075.11001)]. Now, as a further gift, we receive Berndt’s unsurpassed perspective on Ramanujan’s number theory.
By comparison with the author’s previous volumes on Ramanujan, this book is modest in size, a mere 187 pages. Nevertheless, I approached it expecting to find a great deal of interest. In the reading I have not been disappointed.
In eleven pages the preface manages to combine an informative mathematical biography of Ramanujan with a brief discussion of both the history and the mathematics of his notebooks and an outline of the book’s contents. These include a discussion of \(q\)-series and Ramanujan’s theta function (chapter 1), congruences for the partition function \(p(n)\) and the multiplicative arithmetic function \(\tau(n)\) (chapter 2), sums of squares and sums of triangular numbers (chapter 3), Ramanujan’s theory of Eisenstein series (chapter 4), the relation between hypergeometric functions and theta functions (chapter 5), the derivation of modular equations (chapter 6) and the Rogers-Ramanujan continued fraction (chapter 7). Each of the chapters ends with Notes section containing historical commentary, alternative proofs, suggestions for further reading and the like.
Some of the material treated in the book can be found in other texts written at roughly the same level, but by no means all of it. In particular, much of the contents of the chapters 5, 6 and 7 has been available only in the primary sources and in far more comprehensive specialized works.
I am especially pleased to note the dedication of chapter 5 to the connection between hypergeometric functions and theta functions. This link, indeed (more generally) the link between modular forms and hypergeometric functions, has long been observed and studied by number theorists, but the connection is still far from being completely understood. Thus Berndt performs a public service in once more bringing this important relationship to the attention of mathematicians.
These remarks apply equally to chapters 6 and 7, which, like chapter 5, contain much mathematics available until now only in sources exclusively for specialists. In chapter 6 the author brings to bear the power of the main theorem of chapter 5 (Theorem 5.2.8), showing how it leads to the derivation of modular equations (§6.3). Here, Berndt makes the unusual concession to the neophyte of giving his working definition of “modular equation”, as opposed to the more usual practice of definition by example. The significance of the methods discussed in this chapter is suggested by the author’s observation that “there is no single method one can use to discover or construct modular equations. One needs to be resourceful and use a variety of tools” (pp. 140–141).
The final chapter 7 deals exclusively with the Rogers-Ramanujan continued fraction \(R(q)\), “perhaps the most interesting continued fraction in mathematics”, according to the author (p. 154). This function, which first appeared in print in the same article in which Rogers stated and proved the well-known Rogers-Ramanujan identities [Proc. Lond. Math. Soc. 25, 318–343 (1894; JFM 25.0432.01)], turns out to be a modular form of weight 0D on the Hecke congruence group \(\Gamma_0(5)\), a group of genus \(0\). As such, \(R(q)\) satisfies a variety of modular equations; several (of degrees 2, 3 and 5) are given, without proof, in §7.5. This chapter also contains statements of the Rogers-Ramanujan identities (display (7.3.11)), again without proof.
It is of interest that, without any knowledge of Rogers’ work, Ramanujan rediscovered the Rogers identities, but he found no proof of them until after he came upon the proof Rogers gave in the article cited above. Ramanujan’s name was attached to them after he rescued these important relations from the obscurity into which they had fallen following publication.
As the author tells us in his preface, the book presents a carefully selected, “very small fraction of Ramanujan’s work on theta functions and \(q\)-series and their connections with number theory”. Consequently, it provides a useful introduction to the large-scale Berndt and Andrews/Berndt works on the Ramanujan Notebooks. For the relatively inexperienced, including graduate students and suitably prepared undergraduates, the book can likewise be read with the less ambitious goal in mind of learning beautiful and important number theory in our curricula.
Throughout Berndt’s book is informed by the author’s love of his subject, his years of immersion in it and his unmatched status as a Ramanujan scholar,“Number theory in the spirit of Ramanujan” is a gem hat deserves a place in every mathematician’s library.

MSC:

11-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to number theory
11P81 Elementary theory of partitions
11P83 Partitions; congruences and congruential restrictions
11F20 Dedekind eta function, Dedekind sums
11F27 Theta series; Weil representation; theta correspondences
11A55 Continued fractions
33E05 Elliptic functions and integrals
33C75 Elliptic integrals as hypergeometric functions
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Online Encyclopedia of Integer Sequences:

a(n) is the number of partitions of n (the partition numbers).