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4.5
The unfortunate title of this book will probably lead many students of number theory to miss it (as the lack of reviews here show), when in fact, it is THE up to date reference on sieves by the two leading experts in the field!Sieves fall under the specific field of number theory (analysis of sifted sets, such as primes and solving their factors), and the more general field of approximation. With astonishing genius, Euler applied approximation techniques to the numerical solution of intractable differential equations by replacing derivatives with differences and continuous variables with discrete variables, and with the concession of an error term, presciently prepared for high powered computing solutions that couldn't even be imagined in 1750.If that computational leap was prescient, imagine the sieve of Eratosthenes, from the 3rd Century BC! For many years, it was thought that sieve techniques would solve the millennial (million dollar) problem of proving the Riemann Hypothesis. Unfortunately, the parity problem of sieve theory itself should probably be a millennium prize-- no one has yet figured out why sieve theory can't distinguish between odd and even prime factors! So, if you're using your supercomputer to crunch primes with sieves, that zeta function will freeze up when your calculations require that distinction, and so far, sieves have a lot of value, but haven't yet taken a bite out of the million dollar apple.If you want a great book on sieves right up to the early 80's at a fraction of the price of this definitive text, check out: Sieve Methods (Dover Books on Mathematics). For $18 it is astonishing, in that similar reviews are going for over $200 elsewhere.Other than number theory and numerical analysis of prime factorization, where does sieve theory fit in any practical or applied world? As long as DOD and NSA aren't listening, sieve theory is one of the up and coming cornerstones of cryptanalysis and encryption, despite its ancient roots. In fact, they are one of the "surviving" research methods that look intractable to quantum computing, so far.When we marvel at how deep the insights of the Ancients were about numerical methods, it is easy to forget that, WITHOUT computers, this is still one of the few methods they had to crack insurmountable things like non linear ODE's, so it is ironic that the methods have been suddenly reborn WITH brute force computing. Back then, they got "close enough" then checked to see if the solution worked. Today, we do the same thing in estimating the ODE's for airplane wing design, the formulas for which have no explicit solution. If you're in a graduate math program, this book is a must. If you're more into crypto, etc. you might check out the Sieve Methods book above to take less of a bite out of your budget.
