Smooth Numbers

It is explained why smooth numbers are important in practice and also in theory. Ramanujan’s claim on numbers that are 3-smooth is given a thorough investigation. The Dickman function \(\rho \) (u) is introduced together with discussions on its asymptotic formula, starting with the work of de Bruijn, and the more recent contributions from Hildebrand. It is explained how the values of \(\rho \) (u) can be computed accurately with the use of power series. Rankin’s trick is explained with the application of the estimate of the count of smooth numbers. There is an example of the use of smooth numbers to help to solve the discrete logarithm problem.