The Brun-hooley Sieve
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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...
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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 ...
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File LecturesEasyChennai.tex. 1 First lecture: initiation to Brun pure sieve Pure Brun sieve. It was very intricate before the invention of using Rankin’s trick. Multiplicativity. (Brun, 1919a), (Brun, 1919b), (Rankin, 1938), (Murty & Saradha, 1987). The Moebius function is defined by μ(d) = { (−1) when d is a product of t distinct prime factors, 0 otherwise. (1) We have μ(1) = 1. This function...
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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.
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