\documentclass[12pt]{amsart} \usepackage[top=1in, bottom=1in, left=1in, right=1in]{geometry} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %packages \usepackage{amssymb,amsmath,amsthm} \usepackage{mathrsfs} %to use \mathscr fonts \usepackage{bbm} %use with \mathbbm command to produce double-lined'' letters and numbers \usepackage{enumitem} \usepackage[hyphens]{url} \usepackage{graphicx} \usepackage{color} %\usepackage[small,nohug,heads=vee]{diagrams} %\diagramstyle[labelstyle=\scriptstyle] %\usepackage[nouppercase]{scrpage2} %\pagestyle{scrheadings} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %section numbering \renewcommand{\labelenumi}{(\alph{enumi})} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %theorems etc \theoremstyle{theorem} \newtheorem{thm}{Theorem}[section] \newtheorem{lem}[thm]{Lemma} \newtheorem{prop}[thm]{Proposition} \newtheorem{cor}[thm]{Corollary} \newtheorem{rem}[thm]{Remark} \newtheorem{exam}[thm]{Example} \newtheorem{exer}[thm]{Exercise} \newtheorem{dfn}[thm]{Definition} \newenvironment{sol}{\textit{Solution.}} \theoremstyle{definition} \newtheorem{problem}{Problem} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %styled letters \newcommand{\N}{\mathbb{N}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\Q}{\mathbb{Q}} \newcommand{\R}{\mathbb{R}} \newcommand{\C}{\mathbb{C}} \newcommand{\A}{\mathbb{A}} \newcommand{\T}{\mathbb{T}} \newcommand{\U}{\mathbb{U}} \newcommand{\D}{\mathbb{D}} \newcommand{\F}{\mathbb{F}} \newcommand{\CA}{\mathcal{A}} \newcommand{\CB}{\mathcal{B}} \newcommand{\CC}{\mathcal{C}} \newcommand{\CD}{\mathcal{D}} \newcommand{\CE}{\mathcal{E}} \newcommand{\CF}{\mathcal{F}} \newcommand{\CG}{\mathcal{G}} \newcommand{\CH}{\mathcal{H}} \newcommand{\CI}{\mathcal{I}} \newcommand{\CJ}{\mathcal{J}} \newcommand{\CK}{\mathcal{K}} \newcommand{\CL}{\mathcal{L}} \newcommand{\CM}{\mathcal{M}} \newcommand{\CN}{\mathcal{N}} \newcommand{\CO}{\mathcal{O}} \newcommand{\CP}{\mathcal{P}} \newcommand{\CQ}{\mathcal{Q}} \newcommand{\CR}{\mathcal{R}} \newcommand{\CS}{\mathcal{S}} \newcommand{\CT}{\mathcal{T}} \newcommand{\CU}{\mathcal{U}} \newcommand{\CV}{\mathcal{V}} \newcommand{\CW}{\mathcal{W}} \newcommand{\CX}{\mathcal{X}} \newcommand{\CY}{\mathcal{Y}} \newcommand{\CZ}{\mathcal{Z}} \newcommand{\M}{\mathfrak{M}} \newcommand{\m}{\mathfrak{m}} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %mathcommands \newcommand{\abs}{\left\lvert #1 \right\rvert} \newcommand{\set}{\left\{ #1 \right\}} \newcommand{\ds}{\displaystyle} \newcommand{\bs}\boldsymbol{} \newcommand{\sub}{ \substack{ #1 } } \newcommand{\ssum}{ \sideset{}{^*}\sum_{ #1 } } \newcommand{\eq}{ \begin{equation} \label{#1}\begin{split} #2 \end{split} \end{equation} } \newcommand{\al}{\begin{align} #1 \end{align} } \newcommand{\als}{\begin{align*} #1 \end{align*} } \newcommand{\nn}{\nonumber \\} \newcommand{\li}{ \text{li} } \newcommand{\dee}{\mathrm{d}} \DeclareMathOperator{\prob}{{\bf Prob}} \DeclareMathOperator{\meas}{meas} \DeclareMathOperator{\var}{Var} \DeclareMathOperator{\rad}{rad} \DeclareMathOperator{\ord}{ord} \DeclareMathOperator{\sgn}{sgn} \newcommand{\fl}{\left\lfloor#1\right\rfloor} \newcommand{\ceil}{\left\lceil#1\right\rceil} \renewcommand{\hat}{\widehat} \renewcommand{\tilde}{\widetilde} \newcommand{\eps}{\varepsilon} \renewcommand{\phi}{\varphi} \renewcommand{\Re}{\textrm{Re}} \renewcommand{\Im}{\textrm{Im}} \newcommand{\leg}{\left( \frac{#1}{#2} \right) } \newcommand{\mtr}{ \left( \begin{matrix} #1 \end{matrix} \right) } \renewcommand{\bar}{\overline{#1}} \renewcommand{\mod}{\,({\rm mod}\,#1)} %\renewcommand{\mod}{{\ifmmode\text{\rm\,(mod\,$#1$)}\else\discretionary{}{}{\hbox{ }}\rm(mod~$#1$)\fi}} %\newsymbol\dnd 232D %alternative command for \nmid \begin{document} %\thispagestyle{empty} \begin{center} {\bf MAT6627 : the distribution of prime numbers, Fall 2016 }\\ Homework 3, due on Friday December 2, 2016 \end{center} \medskip %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{problem}[10 points]\label{sieve} Let $\CP$ be a set of primes. \begin{enumerate} \item Prove that $\sum_{\substack{a\le x \\ p|a\ \Rightarrow\ p\in\CP}}\frac{1}{a}\sum_{\substack{b\le x \\ p|b\ \Rightarrow\ p\notin\CP}}\frac{1}{b}\ge \sum_{n\le x} \frac{1}{n} .$ \item Prove that $\sum_{\substack{n\le x \\ p|n\ \Rightarrow\ p\in\CP}}\frac{1}{n} \asymp \prod_{p\in\CP\cap[1,x]}\left(1+\frac{1}{p}\right) .$ \end{enumerate} \end{problem} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{problem}\label{smooth}[15 points] An integer $n$ is called $y$-smooth if all of its prime factors are $\le y$. Let $\Psi(x,y)$ be the number of $y$-smooth numbers $\in[1,x]$. \begin{enumerate} \item If $y\in[\sqrt{x},x]$ and $u=\log x/\log y$, then show that $\Psi(x,y) = x(1-\log u) + O(x/\log x) .$ [{\it Hint:} Count, instead, $n\le x$ that are not $y$-smooth.] \item Prove that if $x^\epsilon\le y\le x$, then $\Psi(x,y)\gg_\epsilon x$. [{\it Hint} : Write $\{p\le y\}=\CP\cup\CP'$, where $\CP=\{p\le \sqrt{y}\}$ and $\CP'=\{\sqrt{y}0$, which improves the estimate coming from part (a). [{\it Hint} : Use Problem \ref{smooth}(a) above.] \end{enumerate} \end{problem} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{problem}[15 points] Let $q\ge3$. \begin{enumerate} \item Prove that if $\chi_1,\chi_2$ are two distinct real, non-principal characters $\mod q$, then $\max\{L_q(1,\chi_1),L_q(1,\chi_2)\}\gg1$. [{\it Hint:} Theorem 7.1.1] \item Show that if the Brun-Titchmarsch inequality can be improved to $\pi(x;q,a) \le \frac{(2-\epsilon)x}{\phi(q)\log(x/q)} \quad(x\ge 2q,\, (a,q)=1)$ for some fixed $\epsilon>0$, then $L_q(1,\chi)\gg1$ for all real, non-principal characters $\chi\mod q$. \end{enumerate} \end{problem} \end{document}