MATH Seminar
| Title: The Distribution Of The Number Of Prime Factors With Restrictions - Variations Of The Classical Theme |
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| Seminar: Algebra |
| Speaker: Krishna Alladi of University of Florida |
| Contact: David Zureick-Brown, dzb@mathcs.emory.edu |
| Date: 2017-02-28 at 4:00PM |
| Venue: W306 |
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| Abstract: The study of $\nu(n)$ the number of prime factors of $n$ began with Hardy and Ramanujan in 1917 who showed that $\nu(n)$ has normal order $log\,log\,n$ regardless of whether the prime factors are counted singly or with multiplicity. Their ingenious proof of this utilized uniform upper bounds for $N_k(x)$, the number of integers up to $x$ with $\nu(n)=k$. Two major results followed a few decades later - the Erd\"os-Kac theorem on the distribution more generally of additive functions, and the Sathe-Selberg theorems on the asymptotic behavior of $N_k(x)$ as $k$ varies with $x$ - a significant improvement of Landau's asymptotic estimate for $N_k(x)$ for fixed $k$. We shall consider the distribution of the number of prime factors by imposing certain restrictions - such as (i) requiring all prime factors of $n$ to be $ |
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