How many smooth numbers and smooth polynomials are there?

Smooth numbers are integers whose prime factors are all small (smaller than some threshold y). In the 80s they became important outside of pure math, because Pomerance's quadratic sieve algorithm for factoring integers relied on them and on their distribution.

The density of smooth numbers below x can be approximated -- in some range -- using a peculiar function ρ called Dickman's function, which is defined using a delay-differential equation. 

All of the above is also true for smooth polynomials, which are defined similarly and have practical applications.

We'll survey the quadratic sieve, Dickman's function, the approaches to studying smooth numbers and previous works.

We'll then discuss recent results whose proofs rely on relating the number of smooth numbers to the zeta function and its zeros.

Pomerance asked whether ρ always serves as a lower bound for the density of smooth numbers, even outside the range where we know it is a good approximation. We'll sketch the proof of a positive answer to this question under an hypothesis slightly stronger than the Riemann Hypothesis (RH).

This is closely related to a question of Hildebrand. He showed that under RH, ρ is a good approximation for the density of smooth numbers in a wide range of parameters, and asked whether his range is optimal. This is confirmed by similar methods as those that go into Pomerance's question.