Rosser's sieve

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 ...

Sieve methods

Affine Sieve

We establish the main saturation conjecture in [BGS10] connected with executing a Brun sieve in the setting of an orbit of a group of affine linear transformations. This is carried out under the condition that the Zariski closure of the group is Levi-semisimple. It is likely that this condition is also necessary for such saturation to hold.

Sieve Methods Lecture Notes, Part I the Brun-hooley Sieve

A sieve is a technique for bounding the size of a set after the elements with “undesirable properties” (usually of a number theoretic nature) have been removed. The undesirable properties could be divisibility by a prime from a given set, other multiplicative constraints (divisibility by a perfect square for example) or inclusion in a set of residue classes. The methods usually involve some kin...

Sieve Methods Lecture Notes Selberg’s Upper Bound Sieve

1 Basic inequality For any real numbers ρ d satisfying ρ 1 = 1, and for any natural number m, we have d|m µ(d) d|m ρ d 2 = e|m λ e , λ e = [d 1 ,d 2 ]=e ρ d 1 ρ d 2. Our goal is to optimize the choice of (ρ d) d1. Let m = (n, P (z)), multiply by a n and sum over n using (g): say. Minimizing XG + R is quite difficult. However, if we restrict the support of ρ to d √ D, it is relatively easy to mi...