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'Mangoldt summatory function is $O(x)$'
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| Title of object: |
Mangoldt summatory function is $O(x)$ |
| Canonical Name: |
MangoldtSummatoryFunctionIsOx |
| Type: |
Theorem |
| Created on: |
2007-12-28 21:27:12 |
| Modified on: |
2007-12-28 21:27:12 |
| Classification: |
msc:11A41 |
Preamble:
% 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.
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\usepackage{amssymb}
\usepackage{amsmath}
\usepackage{amsfonts}
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%\usepackage{psfrag}
% need this for including graphics (\includegraphics)
%\usepackage{graphicx}
% for neatly defining theorems and propositions
%\usepackage{amsthm}
% making logically defined graphics
%\usepackage{xypic}
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% define commands here
\newtheorem{thm}{Theorem}
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Content:
\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}. |
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