Department of Mathematics and Statistics

Department of Mathematics and Statistics
Department of Mathematics and Statistics

Number Theory Seminar


Tuesday, August 14th, 2018

Time: 2:30-3:20p.m.  Place: Jeffery Hall 422

Speaker: Arpita Kar (Queen's University)

Title: ON A THEOREM OF HARDY AND RAMANUJAN.

Abstract: In 1917, G.H Hardy and S. Ramanujan coined the phrase ``normal order" and proved that $\omega(n)$ has normal order $\log \log n$. (Here, $\omega(n)$ denotes the number of distinct prime factors of $n$.) In other words, they showed that $\omega(n) \approx \log \log n$ for all but $o(x)$ many integers $n \leq x$, as $x \to \infty$. In this talk, we will show that the size of this exceptional set is, in fact $O(\frac{x}{(\log x)^A)})$ for any $A>0$, improving upon the work of Hardy, Ramanujan and Tur\'an.