The distribution of k-free numbers and integers with fixed number of prime factors

Riemann zeta-function;K-free numbers;Primes in arithmetic progressions;Dirichlet L-function

Meng, Xianchang

关键词: Riemann zeta-function;    K-free numbers;    Primes in arithmetic progressions;    Dirichlet L-function;   

Others  :  https://www.ideals.illinois.edu/bitstream/handle/2142/98338/MENG-DISSERTATION-2017.pdf?sequence=1&isAllowed=y

美国|英语

来源: The Illinois Digital Environment for Access to Learning and Scholarship

【 摘 要 】

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 zeta-function, 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 zeta-function. 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 so-called 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 non-residue 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.