DATE: Tuesday, October 6 (2009), at 3:00pm
PLACE: LIT 305
SPEAKER: Krishnaswami Alladi
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TITLE:
On the number of uncancelled elements
in the sieve of Eratosthenes
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ABSTRACT:
Given the list of primes up to x,
the sieve of Eratostnenes is a procedure to
generate primes from x to x2. Equivalently,
in the interval [1,x], if we sieve out the multiples
of all the primes p < sqrt(x), we are left with
the primes. What happens if we sieve out only
the multiples of primes p<y? We are then left with
integers which have fewer than α=logx/logy
prime factors. When α is small, these integers
are called "almost primes". We will discuss a famous
1950 paper of N. G. deBruijn in which the number
of uncancelled elements by sieving [1,x] by primes
up to y is investigated and asymptotic estimates
uniform in y are obtained. We are led to study the
behavior of a certain function of α which satisfies
a difference-differential equation. As α tends to
infinity, this function tends to exp(-γ), where
γ is Euler's constant.
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