# Mangoldt summatory function is $O(x)$

###### Theorem 1

$\psi(x)=O(x)$, in other , $\frac{\psi(x)}{x}$ is bounded.

Proof.

 $\psi(x)=\sum_{1}^{x}\Lambda(n)=\sum_{\begin{subarray}{c}p\text{ prime}\\ p\leq x\end{subarray}}\lfloor\log_{p}x\rfloor\ln p=\sum_{\begin{subarray}{c}p% \text{ prime}\\ p\leq x\end{subarray}}\left\lfloor\frac{\ln x}{\ln p}\right\rfloor\ln p=\sum_{% \begin{subarray}{c}p\text{ prime}\\ p\leq\sqrt{x}\end{subarray}}\left\lfloor\frac{\ln x}{\ln p}\right\rfloor\ln p+% \sum_{\begin{subarray}{c}p\text{ prime}\\ \sqrt{x}

since $1\leq\frac{\ln x}{\ln p}<2$ if $p>\sqrt{x}$. Continuing, we have

 $\sum_{\begin{subarray}{c}p\text{ prime}\\ p\leq\sqrt{x}\end{subarray}}\left\lfloor\frac{\ln x}{\ln p}\right\rfloor\ln p+% \sum_{\begin{subarray}{c}p\text{ prime}\\ \sqrt{x}

Note that $\pi(x)\ln x\leq 8x\ln 2$ by Chebyshev’s bounds on $\pi(x)$ (http://planetmath.org/BoundsOnPin).

Title Mangoldt summatory function is $O(x)$ MangoldtSummatoryFunctionIsOx 2013-03-22 17:42:59 2013-03-22 17:42:59 rm50 (10146) rm50 (10146) 5 rm50 (10146) Theorem msc 11A41