Analytic Number Theory. Henryk Iwaniec and Emmanuel Kowalski. Author addresses: Rutgers University. E-mail address: [email protected] ETH Zrich. Read Analytic Number Theory: 53 (Colloquium Publications) book reviews & author details and by Henryk Iwaniec (Author), Emmanuel Kowalski (Author). Read Analytic Number Theory book reviews & author details and more at I find it hard to understand proofs by Iwaniec and Kowalski, and Iwaniec in particular.

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Archive ouverte HAL – Analytic Number Theory

Analytic Number Theory distinguishes itself by the variety of tools it uses to establish results. One of the primary attractions of this theory is its vast diversity of concepts and methods. The main goals of this book are to show the scope of the theory, both in classical and modern directions, and to exhibit its wealth and prospects, beautiful theorems, and powerful techniques. The book is written with graduate students in mind, and the authors nicely balance clarity, completeness, and generality.

The exercises in each section serve dual purposes, some intended to improve readers’ understanding of the subject and others providing additional information.

Formal prerequisites for the major part of the book do not go beyond calculus, complex analysis, integration, and Fourier series and integrals. In later chapters automorphic forms become important, with much of the necessary information about them included in two survey chapters. Read more Read less. Applicable only on ATM card, debit card or credit card orders. Cashback will be credited as Amazon Pay balance within 10 days.


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American Mathematical Society; New ed.

Analytic Number Theory

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AMS :: Iwaniec and Kowalski: Analytic Number Theory

Most helpful customer reviews on Amazon. One of the bible in analytic number theory. A good reference book for research. This is the most comprehensive book on analytic number theory that exists. It shows off all the machinery like character sums, Gauss sums, exponential sums, Kloosterman sums, sieves, Dirichlet L-functions and automorphic L-functions, Vinogradov’s method, classical modular forms, theta functions, the spectral theory of automorphic forms, the circle method, equidistribution, and class numbers.

This is a bad book to learn from: Iwaniec and Kowalski are particularly good at using tools from harmonic analysis, rather than only complex analysis as Davenport does; things like bump functions do not appear in Davenport for example.

I find it hard to understand proofs by Iwaniec and Kowalski, and Iwaniec in particular is not a good expositor; in fact, Kowalski seems like a decent expositor and I imagine that this book is written much better than it would have been if Iwaniec were the sole author. The authors do not make much effort to make sure that all the ideas used in a theorem statement or proof have already been introduced to the reader and digested. But this is the best book to get a high level sense of what is going on, and then one can read proofs in other books.


This belongs in any number theorist’s shelf. Preferrably, get two, one for the office and one to keep at home. At the same time, I often recommend it to students as a second book. I’d say that, right now, Montgomery-Vaughan is the user’s guide to classical multiplicative analytic number theory, and Iwaniec-Kowalski is the reference manual with a much broader scope.

Iwaniec and Kowalski often aim for statements of great generality; this makes the life of a working mathematician easier, in that you can often pick a result of theirs “as is”, without needing to reprove things to adapt them to your situation, but it also means that the statements are made harder for a student to absorb. The authors especially the first-named author, on whose course notes the book is based are some of the best people in the field, and the book contains a wealth of information, but the organization is, um, non-existent — results in section a.

So, if you are learning the field, DO get it, but don’t try too hard to read it. This book covers an amazing amount of material and is an extremely good reference, going from more classical topics to spectral methods of automorphic forms and even mentioning the GUE hypothesis and the distribution of zeroes of L-functions.

But this is not a book for average beginners or for self-study, as many of the proofs contain next to no detail. Get to Know Us.

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