DATE: Tuesday, October 13 (2009), at 3:00pm
PLACE: LIT 305
SPEAKER: Krishnaswami Alladi
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TITLE:
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On the number of positive integers < x and free of prime factors > y
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ABSTRACT:
Let ψ(x,y) denote the number of
positive integers <x and free of prime factors > y.
Estimating ψ(x,y) is the conjugate problem
to the estimation of the number of uncancelled
elements in the sieve of Eratosthenes. We will
discuss another classic paper of N. G. deBruijn
(1951) in which uniform estimates for ψ(x,y)
are obtained by analysing a function ρ(α)
where α=log x/log y; ρ(α) is the
famous Dickman function and it satisfies a
difference-differential equation in α). The
study of ψ(x,y) involves a variety of analytic
techniques and tools - the saddle point method,
Dirichlet series, and the prime number theorem.
The function ψ(x,y) occurs in many different
settings ranging from tests for primality to the
distribution of quadratic non-residues.
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