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Title:  The distribution of kfree numbers and integers with fixed number of prime factors 
Author(s):  Meng, Xianchang 
Director of Research:  Ford, Kevin 
Doctoral Committee Chair(s):  Hildebrand, A. J. 
Doctoral Committee Member(s):  Berndt, Bruce C.; Li, Xiaochun 
Department / Program:  Mathematics 
Discipline:  Mathematics 
Degree Granting Institution:  University of Illinois at UrbanaChampaign 
Degree:  Ph.D. 
Genre:  Dissertation 
Subject(s):  Riemann zetafunction
Kfree numbers Primes in arithmetic progressions Dirichlet Lfunction 
Abstract:  This thesis includes four chapters. In Chapter 1, we briefly introduce the history and the main results of the topics of this thesis: the distribution of $k$free numbers and the derivative of the Riemann zetafunction, the generalization of Chebyshev's bias to products of any $k\geq 1$ primes, and the distribution of integers with prime factors from specific arithmetic progressions. In Chapter 2, for any $k\geq 2$, we study the distribution of $k$free numbers. It is known that the number of $k$free numbers up to $x$ is $\widetilde{M}_k(x)\sim \frac{x}{\zeta(k)}$, where $\zeta(s)$ is the Riemann zetafunction. In this chapter, we focus on the distribution of the error term $M_k(x):=\widetilde{M}_k(x)\frac{x}{\zeta(k)}$. Under the Riemann Hypothesis, we prove an equivalent relation between a mean square of the error term $M_k(x)$ and the negative moments of $\zeta'(\rho)$ as $\rho$ runs over the zeros of $\zeta(s)$. Under some reasonable conjectures, we show that $M_k(x)\ll x^{\frac{1}{2k}}(\log x)^{\frac{1}{2}\frac{1}{2k}+\epsilon}$ for all $\epsilon>0$ except on a set of finite logarithmic measure, and that $e^{\frac{y}{2k}}M_k(e^y)$ has a limiting distribution. Finally, based on the analysis of the tail of the limiting distribution, we make a precise conjecture on the maximal order of the error term. In Chapter 3, we generalize the Chebyshev's bias and the socalled prime race problems to the distribution of products of any $k\geq 1$ primes in different arithmetic progressions. For any $k\geq 1$, we derive a formula for the difference between the number of integers $n\leq x$ with $\omega(n)=k$ or $\Omega(n)=k$ in two different arithmetic progressions, where $\omega(n)$ is the number of distinct prime factors of $n$ and $\Omega(n)$ is the number of prime factors of $n$ counted with multiplicity. Under the extended Riemann Hypothesis (ERH) and the Linear Independence Conjecture (LI) for Dirichlet $L$functions, we show that, if $k$ is odd, the integers with $\Omega(n)=k$ have preference for quadratic nonresidue classes; and if $k$ is even, such integers have preference for quadratic residue classes. This result confirms a conjecture of Richard Hudson. However, the integers with $\omega(n)=k$ always have preference for quadratic residue classes. Moreover, as $k$ increases, the biases decrease for both cases. For large $k$, we also give asymptotic formulas for the logarithmic densities of the sets on which the corresponding difference functions have a given sign. In Chapter 4, we prove an asymptotic formula for the number of integers $\leq x$ which can be written as the product of $k ~(\geq 2)$ distinct primes $p_1\cdots p_k$ with each prime factor from a fixed arithmetic progression $p_j\equiv a_j \bmod q$, $(a_j, q)=1$ $(q \geq 3, 1\leq j\leq k)$. For any $A> 0$, our result is uniform for $2\leq k\leq A\log\log x$. Moreover, we show that, there are large biases toward certain arithmetic progressions $\boldsymbol{a}=(a_1, \cdots, a_k)$, and that such biases have connections with Mertens' theorem and the least prime in arithmetic progressions. Unlike the previous two topics, all results in this chapter are unconditional. 
Issue Date:  20170707 
Type:  Text 
URI:  http://hdl.handle.net/2142/98338 
Rights Information:  Copyright 2017 Xianchang Meng 
Date Available in IDEALS:  20170929 
Date Deposited:  201708 
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Dissertations and Theses  Mathematics

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