MangoldtSummatoryFunctionIsOx 2007-12-28 21:27:12 2008-05-24 10:09:27 Theorem Mangoldt summatory function % this is the default PlanetMath preamble. as your knowledge % of TeX increases, you will probably want to edit this, but % it should be fine as is for beginners. % almost certainly you want these \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} % used for TeXing text within eps files %\usepackage{psfrag} % need this for including graphics (\includegraphics) %\usepackage{graphicx} % for neatly defining theorems and propositions %\usepackage{amsthm} % making logically defined graphics %\usepackage{xypic} % there are many more packages, add them here as you need them % define commands here \newtheorem{thm}{Theorem} \begin{thm} $\psi(x)=O(x)$, in other \PMlinkescapetext{words}, $\frac{\psi(x)}{x}$ is bounded. \end{thm} \textbf{Proof. } \[\psi(x)=\sum_1^x \Lambda(n)= \sum_{\substack{p\text{ prime}\\p\leq x}}\lfloor \log_p x\rfloor \ln p= \sum_{\substack{p\text{ prime}\\p\leq x}}\left\lfloor\frac{\ln x}{\ln p}\right\rfloor\ln p= \sum_{\substack{p\text{ prime}\\p\leq \sqrt{x}}}\left\lfloor\frac{\ln x}{\ln p}\right\rfloor\ln p + \sum_{\substack{p\text{ prime}\\\sqrt{x}<p\leq x}}\ln p\] since $1\leq \frac{\ln x}{\ln p}<2$ if $p>\sqrt{x}$. Continuing, we have \[\sum_{\substack{p\text{ prime}\\p\leq \sqrt{x}}}\left\lfloor\frac{\ln x}{\ln p}\right\rfloor\ln p + \sum_{\substack{p\text{ prime}\\\sqrt{x}<p\leq x}}\ln p \leq \sqrt{x}\ln x+\pi(x)\ln x\leq\sqrt{x}\ln x+8x\ln 2=O(x) \] Note that $\pi(x)\ln x\leq 8x\ln 2$ by Chebyshev's \PMlinkname{bounds on $\pi(x)$}{BoundsOnPin}.