Springer Monographs in Mathematics. Berlin: Springer. xii, 201 p. DM 129.00; £44.50; $ 64.95 (2001).
Motivated by an analogy between Nevanlinna’s theory and diophantine approximation, discovered independently by C. F. Osgood and P. Vojta, S. Lang recognized that the careful study of the error term in Nevanlinna’s Second Main Theorem would be of interest in itself. Since then, a flurry of activity in the investigation of the error terms has been developed, due to works of S. Lang, P. M. Wong, and the authors of this book. In particular, the second author of this book obtains many results in this direction. This book collects together the existing work on error terms. It gives an introduction to Nevanlinna’s theory of meromorphic functions, with an emphasis on error terms. It is known that Nevanlinna’s theory of meromorphic functions basically contains two key results, namely First and Second Main theorems. The First Main Theorem is just the re-formulation of Jensen’s formula for meromorphic functions. The Second Main Theorem, however, is an elegant, deep theorem which is the central heart of Nevanlinna’s theory. There are two fundamental different proofs of the Second Main Theorem: One is Nevanlinna’s original proof, based on the so-called logarithmic derivative lemma; and another is due to Ahlfors, based on “negative curvature”. The Second Main theorem with good error term was obtained by P.M. Wong, based on “negative curvature” method, and was obtained by Zhuan Ye(the second author of the book), based on Nevanlinna’s approach, and an earlier work of Gol’dberg-Grinshtein. This book presents nicely the work of Ahlfors-Wong in Chapter 2 and the work of Nevanlinna-Gol’berg-Grinshtein-Ye in Chapter 3 and Chapter 4. I would like, in particular, to mention Chapter 3. Chapter 3 presents the proof of Gol’berg-Grinshtein’s logarithmic derivative lemma. The original proof given by Gol’berg-Grinshtein was short and was difficult to follow. However, the authors incorporate, in this book, various refinements from the work of Gol’berg-Grinshtein, and present the proof of Gol’berg-Grinshtein’s logarithmic derivative lemma beautifully and nicely. Chapter 5 of the book gives some applications. Chapter 6 is called “A Further Digression into Number Theory: Theorems of Roth and Khinchin”. It gives explaining and motivating in some details the connection of Nevanlinna theory and Diophantine approximation. Although this Chapter is short, it is also an important part of the book. The authors have made substantial contributions to the material presented. The book is very well-written. The reviewer highly recommends this book. To close on a personal note, I learned a lot from reading this book.