This is the second edition of a respected text on number theory. The subtitle--an introduction via the density of primes--explains its orientation. It’s certainly ambitious, proving the prime number theorem less than halfway through the book. It’s also thorough, giving many alternative proofs on the infinitude of primes, some of which, especially the topological ones, were new to me.
The foreword gives various possible semester or year-long courses based on parts of the book. If I were teaching such a course, would I use this book? I would certainly have it on the reading list, and at places--for example, the actual proof of the prime number theorem--might rely on it heavily. But would it be my major text? Probably not, for two reasons.
The first is the question of prerequisites. The authors state: “All necessary concepts from abstract algebra and complex analysis are introduced where needed.” I certainly did not find that; much of the algebra just comes as “recall that” and in places is misleading, as when they state the fundamental theorem of abelian groups as analogous to the fundamental theorem of arithmetic, but do not point out that there is only one 22, but more than one group of this size. Much the same is true of analysis; approxeq is introduced before it is defined, and the description of analytic continuation, while sufficient for their purposes, makes no mention of the need for branch cuts, or multivalued equations, or Riemann surfaces.
The second is the number of errors or confusions. Some are historical. The authors state, several times, that the Hadamard and de la Vallée Poussin proofs were the genesis of analytic number theory (1896), but that Dirichlet introduced analytic methods (1837). And Cassels, though undoubtedly a great mathematician, certainly did not invent the abelian group view of elliptic curves. Some are mathematical, or at least pedagogic, as in the start of Lemma 3.1.13 where a couple of lines need to be inserted.
More worrying for me, though this might not concern everyone, are the algorithmic weaknesses. Page 241 states that linear-time checking for k-th powers “can be done,” but gives no reference, and [1] is actually subtler than that. The description of Lenstra’s elliptic curve method is hardly an algorithm at all, and I cannot see how this description leads to their example.
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