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.