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A Course in Analytic Number Theory

Marius Overholt
American Mathematical Society
Publication Date: 
Number of Pages: 
Graduate Studies in Mathematics 160
[Reviewed by
Michael Berg
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This book is pretty much what it advertises itself to be: Overholt states in his Preface that the text is meant for graduate students looking to start in analytic number theory, and it does the job. Accordingly it is pitched at an accessible level, the explicit prerequisites being undergraduate analysis, complex analysis, linear algebra, and abstract algebra — in the present case of analytic number theory, this should be understood in the more old-fashioned sense: not just group theory, but also rings and fields and a decent chunk of Galois theory. After all, number fields are at the heart of the subject. Already in Overholt’s third chapter, “Characters and Euler Products,” we see group representations, Fourier analytic methods, and of course character theory, and this theme is continued in the ninth chapter, where Euler products are considered in the context of the aforementioned number fields (and the chapter closes with a discussion of nothing less than Artin L-functions). This trajectory is already sufficient, I think, to recommend the book: it’s a wonderfully holistic approach to a central theme in analytic number theory; but there is a lot more to it than that. Overholt also covers such things as the Hardy-Littlewood circle method (Chapter 4), the prime number theorem (in Chapter 6: every analytic number theory book should have this in it, of course), and (in Chapter 7) the Siegel-Walfisz Theorem. So, to be sure, we’re in the thick of it.

Additionally, Overholt includes a marvelous (eighth) chapter, titled “Mainly Analysis,” in which he discusses, if I may put it so (with apologies to Oscar Hammerstein), “some of my favorite things.” To wit, we get theta functions (properly in the wake of Poisson summation), the gamma function, Riemann’s ζ-functional equation (and in Chapter 10 Overholt goes on to address the beautiful and mysterious relationship between the distribution of the primes and the Riemann Hypothesis: “There is a precise relationship between the size of the error term in the Prime Number Theorem and the real part of the zeros of the Riemann zeta function,” after which it’s on to the theorem of von Koch). Then comes the functional equation for a Dirichlet L-function attached to a character mod. q (with q at least 3). This set of topics also exemplifies another virtue of the present book, i.e. Overholt’s inclusion of a wealth of historical and literature-oriented remarks; we read on p. 231, for example, that “[t]he standard reference for the analytic theory of the Riemann zeta function is … [still!] Titchmarsh, revised and with end-of-chapter notes by … Heath-Brown … [and] there is also a comprehensive treatise by Aleksandar Ivič … [as well as a text] by S. J. Patterson [that] is strong on explanation and motivation and has many exercises …” Indeed.

Thus, as I said at the outset, I agree that this book is a proper text for a graduate student (with a pretty strong background) keen on getting into analytic number theory, and it’s quite a good one. It’s well-written, rather exhaustive, and well-paced. The choice of themes is good, too, and will form a very sound platform for future studies and work in this gorgeous field. 

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Michael Berg is Professor of Mathematics at Loyola Marymount University in Los Angeles, California.