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Ernie Croot
Ernie Croot
Title: ``Smooth Numbers in Short Intervals''
Abstract: There are several algorithms in computational number
theory,
whose run time analysis require good lower bounds for
$\psi([x,x+4 \sqrt{x}], x^\epsilon)$, which is the number of integers
in the interval $[x,x+4\sqrt{x}]$ having no prime divisor
$>x^\epsilon$.
Unfortunately, even showing that $\psi([x,x+4\sqrt{x}], x^\epsilon) >
0$
for all $\epsilon > 0$ and sufficiently large $x$ seems to be a very
difficult problem, and remains unsolved. In this talk I will give
sketch
of a proof that for all $\delta > 0$, there exists $x_0(\delta)$ such
that if $x > x_0(\delta)$, then
$$
\psi([x,x+c(\delta)\sqrt{x}], x^{3/(14\sqrt{e})+\delta}) >
x/\log^{\log 4+o(1)}x,
$$
where $c(\delta)$ is some constant which depends only on $\delta$.
This result is the sharpest of its kind for intervals of the
form $[x,x+c(\delta)\sqrt{x}]$.
Robert F. Coleman
Last modified: Mon Jan 24 15:19:17 PST 2000