Room 326

Asymptotic Formulae for Smooth Numbers, and Its Implications to Other Counting Problems

Kamalakshya Mahatab

IMSc

In this talk we will have an expedition through the asymptotic formulae, and various techniques, due to many different mathematicians, to investigate smooth numbers. The numbers whose prime factors belongs to an interval [1, y] are called smooth numbers: first studied by Rankin(1938) to give an upper bound; and later improved to asymptotic formulae by de Bruijn(1966), Hildebrand(1986), Tenenbaum(1986), and Sias(1989). We will have a brief sketch of these techniques; and its implications to counts involving various L-functions, and more complicated problems, when [1, y] is replaced by several union of intervals. As tools to these techniques, we will explore the properties of: Dickman's function(\rho(u)), de Bruijn's function(\Lambda(x, y)), and truncated product of Zeta(\zeta(s, y)). In the investigations of Dirichlet L-functions, we will see, how the contribution of Siegel Zero matters to the main term, and to error. We will also explore the use of these techniques, due to Goldston(1988), to sieve out large prime factors.