Sieve Methods

Preface Sieve methods have had a long and fruitful history. The sieve of Eratosthenes (around 3rd century B.C.) was a device to generate prime numbers. Later Legendre used it in his studies of the prime number counting function π(x). Sieve methods bloomed and became a topic of intense investigation after the pioneering work of Viggo Brun (see [Bru16],[Bru19], [Bru22]). Using his formulation of the sieve Brun proved, that the sum ∑ p, p+2 both prime 1 p converges. This was the first result of its kind, regarding the Twin-prime problem. A slew of sieve methods were developed over the years — Selberg's upper bound sieve, Rosser's Sieve, the Large Sieve, the Asymptotic sieve, to name a few. Many beautiful results have been proved using these sieves. The Brun-Titchmarsh theorem and the extremely powerful result of Bombieri are two important examples. Chen's theorem [Che73], namely that there are infinitely many primes p such that p + 2 is a product of at most two primes, is another indication of the power of sieve methods. Sieve methods are of importance even in applied fields of number theory such as Algorithmic Number Theory, and Cryptography. There are many direct applications, for example finding all the prime numbers below a certain bound, or constructing numbers free of large prime factors. There are indirect applications too, for example the running time of several factoring algorithms depends directly on the distribution of smooth numbers in short intervals. The so called undeniable signature schemes require prime numbers of the form 2p + 1 such that p is also prime. Sieve methods can yield valuable clues about these distributions and hence allow us to bound the running times of these algorithms. In this treatise we survey the major sieve methods and their important applications in number theory. We apply sieves to study the distribution of square-free numbers, smooth numbers, and prime numbers. The first chapter is a discussion of the basic sieve formulation of Legendre. We show that the distribution of square-free numbers can be deduced using a square-free sieve 1. We give an account of improvements in the error term of this distribution, using known results regarding the Riemann Zeta function. The second chapter deals with Brun's Combinatorial sieve as presented in the modern language of [HR74]. We apply the general sieve to give a simpler proof of a theorem of Rademacher [Rad24]. The bound obtained by …