Ramanujan’s notebooks. Part IV. (English) Zbl 0785.11001

New York: Springer-Verlag. xii, 451 p. (1994).
[For the reviews of Vols. I–III see Zbl 0555.10001, Zbl 0716.11001, Zbl 0733.11001.]
From the preface: “This book …the fourth of five volumes devoted to the editing of Ramanujan’s notebooks …is the first of two volumes devoted to proving the results found in the unorganized portions of the second …and …third notebook. We shall also prove those results in the first notebook that are not found in the second or third notebooks.”
As the author explains, there are three notebooks being edited in this series. The second is a revision and expansion of the first. Berndt’s first three volumes cover all the material organized by Ramanujan. After Ramanujan had produced this organized opus, he recorded results in an unorganized fashion in over 100 subsequent pages of the second and third notebooks.
Thus Berndt has provided the organization of this material. He has chosen to stick closely to the order of appearance (within each of Berndt’s chapters, of course).
As with previous volumes in this series, Berndt provides a plethora of cross-referencing aids. Correspondences are offered between results in this volume and Ramanujan’s papers, questions he posed in the Journal of the Indian Mathematical Society, and Ramanujan’s collected works. Also, there is a chapter clearly indicating the relation of (each of the) first notebook’s (organized) results with the second notebook.
As I have noted in my previous Zbl review of Volume III (1991), Berndt maintains throughout an extraordinary level of enthusiasm – after nearly 15 years, his awe of Ramanujan and his results is undiminished. This shines through again and again in the text and adds greatly to the readability and enjoyability of this book.
Here are (some of the ten) Chapter titles: Elementary results, Number theory, Ramanujan’s theory of prime numbers, Theta-functions and modular equations, \(q\)-series; Integrals, Special functions, and Partial fraction expansions. As these titles indicate, if one wishes to buy a single volume of the series (so far), just for pure personal enjoyment, this is the one!
Now that Berndt is reaching the end of the project, the scope and majesty of what he has accomplished is becoming clear. He has made a vast amount beautiful and deep mathematics accessible, finally displaying Ramanujan’s full oeuvre. As a result, it will at last become possible to truly assess Ramanujan’s genius.


11-02 Research exposition (monographs, survey articles) pertaining to number theory
11-03 History of number theory
01A75 Collected or selected works; reprintings or translations of classics
01A60 History of mathematics in the 20th century