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Title:  Generalized and restricted multiplication tables of integers 
Author(s):  Koukoulopoulos, Dimitrios 
Director of Research:  Ford, Kevin 
Doctoral Committee Chair(s):  Berndt, Bruce C. 
Doctoral Committee Member(s):  Ford, Kevin; Hildebrand, A.J.; Ahlgren, Scott 
Department / Program:  Mathematics 
Discipline:  Mathematics 
Degree Granting Institution:  University of Illinois at UrbanaChampaign 
Degree:  Ph.D. 
Genre:  Dissertation 
Subject(s):  Divisors
Multiplication Table Problem Shifted Primes 
Abstract:  In 1955 Erdõs posed the multiplication table problem: Given a large integer N, how many distinct products of the form ab with a≤N and b≤N are there? The order of magnitude of the above quantity was determined by Ford. The purpose of this thesis is to study generalizations of Erdõs's question in two different directions. The first one concerns the kdimensional version of the multiplication table problem: for a fixed integer k≥3 and a large parameter N, we establish the order of magnitude of the number of distinct products n_1...n_k with n_i≤N for all 1≤i≤k. The second question we shall discuss is the restricted multiplication table problem. More precisely, for a set of integers B we seek estimates on the number of distinct products ab in B with a≤N and b≤N. This problem is intimately connected with the available information on the distribution of B in arithmetic progressions. We focus on the special and important case when B={p+s:p prime} for some fixed nonzero integer s. Ford established upper bounds of the expected order of magnitude for {p+s=ab:a≤N,b≤N}. We prove the corresponding lower bounds thus determining the size of the quantity in question up to multiplicative constants. 
Issue Date:  20100820 
URI:  http://hdl.handle.net/2142/16757 
Rights Information:  Copyright 2010 Dimitrios Koukoulopoulos 
Date Available in IDEALS:  20100820 
Date Deposited:  201008 
This item appears in the following Collection(s)

Graduate Dissertations and Theses at Illinois
Graduate Theses and Dissertations at Illinois 
Dissertations and Theses  Mathematics