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### Begin Citation ### Do not delete this line ###
%R 2000-10
%U /u1/csegrad/cdx/public_html/Sieve.ps
%A Charles, Denis
%T Sieve Methods
%D July 18, 2000
%I Department of Computer Science and Engineering, SUNY Buffalo
%K Sieves, Bombieri's Theorem, Brun-Titchmarsh Theorem, Squarefree numbers, Distribution of Primes.
%X There are many open problems regarding the distribution of primes, 
for example the {\em Goldbach} conjecture that every even number $ > 2$ 
is a sum of two primes, the {\em Twin prime} problem that there 
are infinitely many prime pairs $(p,p+2)$. Sieve methods were developed 
as a tool to attack these problems and they have become remarkably useful 
in many other situtations. Sieve methods have shown for example that 
there are infinitely many primes $p$ such that $p+2$ is a product of 
at most two prime factors. In our work we show a wide variety of applications 
where Sieve methods give the best known results; in particular we use Sieves 
to study the distribution of {\em Square-free} numbers, {\em Smooth numbers}, 
{\em Quasi-primes}, and primes in short intervals. We also derive Bombieri's theorem 
using the {\em Large sieve} and use this to prove for example that there 
are infinitely many primes $p$ such that $p+k$ is squarefree for any $k > 0$, 
and a product of at most 7 distinct primes.