Complex Analysis with Applications to Number Theory

Tarlok Nath Shorey is an accomplished number theorist with long experience lecturing on complex analysis. This text is adapted from lectures first given at the Tata Institute 45 years ago, and updated over the last 10 years at IIT Bombay. It covers the core material of a first complex analysis course briefly in the first two chapters, with an emphasis on the topological underpinnings that such courses often skim over. The middle chapters introduce a range of advanced topics: the Riemann mapping theorem, harmonic functions and the Dirichlet problem, the Picard theorems, and product expansions. Some, but not all, of these advanced topics are used in the final chapters on analytic number theory. The number theory chapters build toward a proof of the prime number theorem with error term and the prime number theorem in arithmetic progressions. A final chapter on the Baker theorem in transcendence theory connects to Shorey’s research interests but does not fit particularly well with the rest of the material. 

 

The strength of this text is in its concise but complete development of each topic. Difficult theorems are often proven via a string of quick lemmas. For example, chapter 5 neatly proves the Picard theorems in a sequence of steps starting from the Borel-Carathéodory lemma, which is reused in the number theory chapters later. On the other hand, some readers may be frustrated with Shorey’s style, which still feels a lot like a set of lecture notes; especially in the later chapters, it is heavy on equations and light on explication. Some chapters, like chapter 7 on the zeta function, have abundant, well-chosen exercises, but others have a more limited selection.  

 

This book is not appropriate for a first undergraduate or graduate complex analysis course. It assumes a basic familiarity with the properties of analytic functions and does not always present topics in the most intuitive order for a newcomer to the subject. For example, the Cauchy-Riemann equations do not appear until chapter 4, after the full development of complex integration and the Riemann mapping theorem. Many classic complex analysis textbooks cover the fundamentals of the subject more comprehensively. 

 

If an ambitious instructor would like to teach the fundamentals with number-theoretic motivation and applications, they would still likely find this text too broad. Stein and Shakarchi's volume on complex analysis might be a better alternative for such a course. Most new complex analysis texts take a more streamlined approach to the prime number theorem than Shorey's. You may not get an error term, but you get a shorter, self-contained proof that showcases contour integration techniques. 

 

On the other hand, for a course primarily on analytic number theory, this book does not offer quite enough material. There are many excellent textbooks that focus solely on the analysis of zeta and L-functions, and get the reader closer to current research on the subject.  Shorey’s text is best used for a second complex analysis course covering a range of advanced topics, or as supplemental reading.

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Ian Whitehead is a number theorist specializing in automorphic forms and connections between number theory and Kac-Moody theory. He teaches at Swarthmore College.