Ofir Gorodetsky: How many smooth numbers and smooth polynomials are there? (NTWS 164)
Channel: Number Theory Web Seminar
Date: 2023-04-16

Abstract: Smooth numbers are integers whose prime factors are smaller than a threshold y. In the 80s they became important outside of pure math, as Pomerance's quadratic sieve for factoring integers relied on their distribution. The density of smooth numbers up to x can be approximated, in some range, using a peculiar function ρ called Dickman's function, defined via a delay-differential equation. All of the above is also true for smooth polynomials over finite fields.
We'll survey these topics and discuss recent results concerning the range of validity of the approximation of the density of smooth numbers by ρ, whose proofs rely on relating the counting function of smooth numbers to the Riemann zeta function and the counting function of primes. In particular, we uncover phase transitions in the behavior of the density at the points y=(log x)^2 (as conjectured by Hildebrand) and y=(log x)^(3/2), when previously only a transition at y=log x was known and understood. These transitions also occur in the polynomial setting. We'll also show that a standard conjecture on the error in the Prime Number Theorem implies ρ is always a lower bound for the density, addressing a conjecture of Pomerance.
Link to recording


Number Theory Web Seminar: https://www.ntwebseminar.org

Original air date:
Thursday, March 23, 2023 (9am PDT, 12pm EDT, 4pm GMT, 5pm CET, 6pm Israel Standard Time, 9:30pm Indian Standard Time)
Friday, March 24, 2023 (12am CST, 3am AEDT, 5am NZDT)