## On the local distribution of the number of small prime factors

#### When/Where:

February 16, 2016, 3:00 — 3:50 pm at LIT 368.

#### Abstract:

In the second lecture, I discussed the my work on the distribution of the number of prime factors among integers all of whose prime factors are large, or all are small. With that as a background, I will discuss in this third lecture the local distribution of $$\nu_y(n)$$, the number of prime factors of $$n$$ which are less than $$y$$. This leads to Todd Molnar’s thesis. More precisely, I will consider asymptotic estimates for $$N_k(x,y)$$, the number of integers $$\le x$$ for which $$\nu_y(n)=k$$, as $$y$$ varies with $$x$$ and $$k$$ varies as well. When $$\frac{\log x}{\log y} >1$$, the behavior of $$N_k(x,y)$$ is strikingly different from the classical case, but as k tends to $$\log\log y$$, the mean, the behavior is as in the classical case. To study this problem, we investigate sums involving $$z^{\nu_y(n)}$$, where $$z$$ is a complex number. Previously,I had investigated such sums when $$0<z<1$$ using sieve methods. The investigation for complex $$z$$ can be done using the analytic method of Selberg when $$y$$ is small, and by the use of difference-differential equations when $$y$$ is large. The interplay of a variety of techniques is fascinating.