The number of prime factors function and integers with restricted prime factors
February 2, 2016, 3:00 — 3:50 pm at LIT 368.
The study of the distribution of of values of the number of prime factors of the integers goes back to Hardy-Ramanujan. Then with the work of Turan and Erdos-Kac, Probabilistic Number Theory was born. Selberg showed how analytic methods could be used to study the “local” distribution of the number of prime factors.
The distribution of integers all of whose prime facts are small, or all are large, is of fundamental importance. N. G. deBruijn obtained strong uniform estimates for these two problems. My work thirty years ago involved a study of the distribution of the number of prime factors among integers all of whose prime factors are large, or all are small. In this first of two talks, I shall review all of the above mentioned results as a background for recent work on the number of restricted prime factors that reveals some surprising variations on the classical theme. These recent observations that lead to the thesis of Todd Molnar, will be the theme of the second lecture.