An extension of the Dickman function and its application

Chaogui Zhang, Purdue University

Abstract

In this thesis, we study a generalization of the Dickman function and its applications. We first generalize the concept of a smooth integer to make it suitable to analyze the Large-Prime Variations of the Quadratic Sieve(QS) and Number Field Sieve (NFS). Smooth integers are the ones whose largest prime divisors are bounded. In our generalization, we will bound the largest prime divisor and the (k + 1)st largest prime divisor and ignore the size of the other prime divisors in between. Such integers will be called k-semismooth. A heuristic argument will first be given to derive recurrence formulae for the asymptotic distribution of k-semismooth integers. Using that, we define a generalization of the Dickman function. Then we give a rigorous proof for the recurrence formulae, and explore some properties of the generalized Dickman function. We also present a method for computing this function. Numerical results and applications to the MPQS and NFS will then be discussed. At the end, we will investigate the smooth integer distribution in a short interval, and use results available in this area to get an estimate of the function f(n) defined as the smallest positive integer z such that the interval [n, n + z] contains a set of integers (including n and n + z) whose product is a perfect square. This problem was proposed by Selfridge and Meyerowitz.

Degree

Ph.D.

Advisors

Wagstaff, Purdue University.

Subject Area

Mathematics

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