##
**Ramanujan’s notebooks. Part V.**
*(English)*
Zbl 0886.11001

New York, NY: Springer. xiii, 624 p. (1998).

In the time between now and when the reviewer’s Zbl entry for volume IV in this set was written, the entire effort has won the coveted 1996 American Mathematical Society’s Steele Prize for Exposition. Here is the citation from the NAMS November 1996 issue: “To Bruce C. Berndt for the four volumes Ramanujan’s notebooks, Parts I, II, III, and IV [Springer (1985; Zbl 0555.10001), (1989; Zbl 0716.11001), (1991; Zbl 0733.11001) and (1994; Zbl 0785.11001)]. In recognition of Berndt’s heroic and extraordinary achievement in exposing to the general mathematical researcher a trove of results that were utterly inaccessible before, the AMS decided this year, exceptionally, to broaden the standard interpretation of “exposition”. In an impressive scholarly accomplishment spread out over 20 years, Berndt has provided a readable and complete account of the notebooks, making them accessible to other mathematicians. Ramanujan’s enigmatic, unproved formulas are now readily available, together with context and explication, often after the most intense and clever research efforts on Berndt’s part.” What more can I say:

We turn to our review of the current volume. The following text is from the author’s preface: “This book is the fifth and final volume devoted to the editing of Ramanujan’s notebooks. Parts I–III, published, respectively, in 1985, 1989, and 1991, contain accounts of Chapters 1-21 in the second notebook, a revised enlarged edition of the first. Part IV, published in 1994, contains results from the 100 unorganized pages in the second notebook and the 33 unorganized pages comprising the third notebook. Also examined in Part IV are the 16 organized chapters in the first notebook, which contain very little that is not found in the second notebook. In this fifth volume, we examine the remaining contents from the 133 unorganized pages in the second and third notebooks, and the claims in the 198 unorganized pages of the first notebook that cannot be found in the succeeding notebooks. In contrast to the organized portion of the first notebook, the unorganized material in the first notebook contains several results, particularly about class invariants, singular moduli, and values of theta-functions, which are not recorded in the second and third notebooks.”

Here are the chapter titles: 32 Continued fractions, 33 Ramanujan’s theories of elliptic functions to alternative bases, 34 Class invariants and singular moduli [the reviewer will discuss this spectacular chapter below], 35 Values of theta-functions, 36 Modular equations and theta-function identities in notebook I, 37 Infinite series, 38 Approximations and asymptotic expansions, 39 Miscellaneous results in the first notebook.

The reviewer took the liberty of contacting [e-mail; February, 1998] the author to get his advice on the following question: With the five volumes in hand, how would one ascertain what Ramanujan’s notebooks contain on topic \(x\)? Here is a digest of Professor Berndt’s reply: “To check [what] Ramanujan did [on topic \(x\)], [first consult appropriate chapters and the indices of Volumes] I–III [as these] cover Ramanujan’s organized work in the second notebook. Thus, looking at chapter headings to find if Ramanujan proved a theorem on topic \(x\) would be the first thing to do. In the vast amount of unorganized material [volumes IV and V], there are both new topics and old topics. In Parts IV, V I have tried to organize things as [well as] possible.”

Also of note is the volume Ramanujan: Letters and commentary [B. C. Berndt and R. A. Rankin, History Math. 9, AMS, Providence (1995; Zbl 0842.01026)]. From this the reader will glean great insight into Ramanujan’s personal life and history, especially after his “discovery” by Hardy.

Let me close with the suggestion that no reader should fail to examine Chapter 34 of the present volume, Class invariants and singular moduli. But before doing so, I suggest a look at the modern “Introduction to the construction of class fields” [H. Cohn, Cambridge Stud. Adv. Math. 6, Cambridge University Press (1985; Zbl 0571.12001)]. Here we see, on page 3, a restatement of Weber’s famous result: “All ring class fields can be constructed explicitly by adjoining to the splitting field the values of certain…modular functions…” (Emphasis in the original). Much later (page 179) Cohn says, “The problem remains to evaluate [the modular invariant function] \(j\) with the foreboding that all but the smallest values will be beyond human comprehension in magnitude!” The upshot is to replace these values of \(j\) by the class invariants (algebraic numbers) of Chapter 34. Berndt tabulates (Section 2) all such invariants given in the Ramanujan notebooks – and, in subsequent sections of the chapter, derives them with a variety of deep analytic methods. To paraphrase Cohn, these class invariant formuli possess charm that is, perhaps, at the limit of human comprehension.

We turn to our review of the current volume. The following text is from the author’s preface: “This book is the fifth and final volume devoted to the editing of Ramanujan’s notebooks. Parts I–III, published, respectively, in 1985, 1989, and 1991, contain accounts of Chapters 1-21 in the second notebook, a revised enlarged edition of the first. Part IV, published in 1994, contains results from the 100 unorganized pages in the second notebook and the 33 unorganized pages comprising the third notebook. Also examined in Part IV are the 16 organized chapters in the first notebook, which contain very little that is not found in the second notebook. In this fifth volume, we examine the remaining contents from the 133 unorganized pages in the second and third notebooks, and the claims in the 198 unorganized pages of the first notebook that cannot be found in the succeeding notebooks. In contrast to the organized portion of the first notebook, the unorganized material in the first notebook contains several results, particularly about class invariants, singular moduli, and values of theta-functions, which are not recorded in the second and third notebooks.”

Here are the chapter titles: 32 Continued fractions, 33 Ramanujan’s theories of elliptic functions to alternative bases, 34 Class invariants and singular moduli [the reviewer will discuss this spectacular chapter below], 35 Values of theta-functions, 36 Modular equations and theta-function identities in notebook I, 37 Infinite series, 38 Approximations and asymptotic expansions, 39 Miscellaneous results in the first notebook.

The reviewer took the liberty of contacting [e-mail; February, 1998] the author to get his advice on the following question: With the five volumes in hand, how would one ascertain what Ramanujan’s notebooks contain on topic \(x\)? Here is a digest of Professor Berndt’s reply: “To check [what] Ramanujan did [on topic \(x\)], [first consult appropriate chapters and the indices of Volumes] I–III [as these] cover Ramanujan’s organized work in the second notebook. Thus, looking at chapter headings to find if Ramanujan proved a theorem on topic \(x\) would be the first thing to do. In the vast amount of unorganized material [volumes IV and V], there are both new topics and old topics. In Parts IV, V I have tried to organize things as [well as] possible.”

Also of note is the volume Ramanujan: Letters and commentary [B. C. Berndt and R. A. Rankin, History Math. 9, AMS, Providence (1995; Zbl 0842.01026)]. From this the reader will glean great insight into Ramanujan’s personal life and history, especially after his “discovery” by Hardy.

Let me close with the suggestion that no reader should fail to examine Chapter 34 of the present volume, Class invariants and singular moduli. But before doing so, I suggest a look at the modern “Introduction to the construction of class fields” [H. Cohn, Cambridge Stud. Adv. Math. 6, Cambridge University Press (1985; Zbl 0571.12001)]. Here we see, on page 3, a restatement of Weber’s famous result: “All ring class fields can be constructed explicitly by adjoining to the splitting field the values of certain…modular functions…” (Emphasis in the original). Much later (page 179) Cohn says, “The problem remains to evaluate [the modular invariant function] \(j\) with the foreboding that all but the smallest values will be beyond human comprehension in magnitude!” The upshot is to replace these values of \(j\) by the class invariants (algebraic numbers) of Chapter 34. Berndt tabulates (Section 2) all such invariants given in the Ramanujan notebooks – and, in subsequent sections of the chapter, derives them with a variety of deep analytic methods. To paraphrase Cohn, these class invariant formuli possess charm that is, perhaps, at the limit of human comprehension.

Reviewer: M.Sheingorn (New York)

### MSC:

11-02 | Research exposition (monographs, survey articles) pertaining to number theory |

11-03 | History of number theory |

33-02 | Research exposition (monographs, survey articles) pertaining to special functions |

33-03 | History of special functions |

33E05 | Elliptic functions and integrals |

01A75 | Collected or selected works; reprintings or translations of classics |

01A60 | History of mathematics in the 20th century |

41-03 | History of approximations and expansions |

41A58 | Series expansions (e.g., Taylor, Lidstone series, but not Fourier series) |

41A60 | Asymptotic approximations, asymptotic expansions (steepest descent, etc.) |