Sums of Multiplicative Functions Over Integers Without Large Prime Factors and Related Differential Difference Equations

Abstract

Integers without large prime factors arise naturally in various areas of number theory. Counting functions of such integers have been the subject of numerous past studies. Traditionally, Psi(x, y) denotes the number of positive integers up to x, all of whose prime divisors are less than or equal to y. The estimate of Psi(x, y) is given in terms of Dickman's famous function rho(u), where u = log x/log y. In this thesis, we study generalizations of Psi(x, y): Let S(x, y) denote the set of positive integers up to x, all of whose prime divisors are at most y, and we consider sums M( x, y) = n∈Sx,y h(n), and m( x, y) = n∈Sx,y h(n)/n, where h(n) is a non-negative multiplicative function satisfying a set of weak summatory conditions on primes and prime powers. When h(n) ≡ 1 we have M( x, y) = Psi(x, y). The estimates of M(x, y) and m(x, y) are expressed in terms of a family of continuous functions that are related to Dickman's function. We obtain these results by an inductive argument that was inspired by A. Hildebrand's estimate of Psi(x, y), starting with initial estimates of m(x, y) and M(x, y) of H. Halberstam.