### Smooth numbers and polynomials, and the Dickman function

Heilbronn Number Theory Seminar

19th October 2022, 4:00 pm – 5:00 pm

Fry Building, 2.04

An integer is y-smooth (or y-friable) if all its prime factors are at most y in size. Such numbers play an important role in computational number theory. A similar definition, with similar applications, applies for smooth polynomials over a finite field.

It has been known for more than 90 years that the densities of smooth numbers and polynomials grow (in certain parameter ranges) like the Dickman function, which will be defined in the talk.

I will survey the tools for studying smooth numbers and polynomials. I'll explain my work (in progress), which studies the relationship between smooth numbers/polynomials and the Dickman function by introducing a new approximation for the number of smooth numbers/polynomials.

In particular, we make progress, conditionally on the Riemann Hypothesis, on the following question of Pomerance: Is the density of smooth numbers always larger than the Dickman function? This turns out to relate, in a subtle way, to the error term arising when counting prime numbers.

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