December 31, 2018 - January 4, 2019
Public Access Category: All |
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- Time:
- 3:30pm
- Location:
- Ramanujan Hall
- Description:
- SPEAKER: Krishnaswami Alladi
AFFILIATION: University of Florida
TITLE: "On the local distribution of the number of small prime
factors - a variation of the classical theme"
DAY & DATE: Thursday, 3rd January 2018.
TIME: 3.30 PM.
VENUE: Ramanujan Hall.
ABSTRACT: The global distribution of $\nu_y(n)$, the number
of (distinct) prime factors of $n$ which are $
role in the proof of the celebrated Erd\"os -Kac theorem on the
distribution of $\nu(n)$, the number of distinct prime factors
of $n$. Although much is known about the "local distribution"
of $\nu(n)$, namely the asymptotics of the function $N_k(x)=
\sum_{n\le x, \nu(n)=k}1$ (Landau-Sathe-Selberg), little attention
has been paid to the local distribution of $\nu_y(n)$. In discussing
the asymptotic behavior of $N_k(x,y)=\sum_n\le x, \nu_y(n)=k)1$,
we noticed a very interesting variation of the classical theme that
seems to have escaped attention. To explain this phenomenon,
we will obtain uniform asymptotic estimates for $N_k(x,y)$ by a variety of
analytic techniques such as those of Selberg, and of Buchstab-De Bruijn
(involving difference-differential equations). This is joint work with my
recent PhD student Todd Molnar. The talk will be accessible to
non-experts.