PlanetMath: Mangoldt summatory function is $O(x)$

(more info)

Math for the people, by the people.

donor list - find out how

Main Menu

sections Encyclopædia
Papers
Books
Expositions

meta Requests (236)
Orphanage
Unclass'd (1)
Unproven (541)
Corrections (47)
Classification

talkback Polls
Forums
Feedback
Bug Reports

downloads Snapshots
PM Book

information News
Docs
Wiki
ChangeLog
TODO List
Copyright
About

Mangoldt summatory function is $O(x)$

(Theorem)

Theorem 1 $\psi(x)=O(x)$ in other words, $\frac{\psi(x)}{x}$ is bounded.

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 <</A>49#>bounds on $\pi(x)$ http://planetmath.org/encyclopedia/BoundsOnPin.html.

"Mangoldt summatory function is $O(x)$

" is owned by rm50.

(view preamble | get metadata)

This object's parent.

Log in to rate this entry.
(view current ratings)

Cross-references: proof, bounded

This is version 2 of Mangoldt summatory function is $O(x)$

, born on 2007-12-28, modified 2008-05-24.
Object id is 10161, canonical name is MangoldtSummatoryFunctionIsOx.
Accessed 668 times total.

Classification:

AMS MSC:

11A41 (Number theory :: Elementary number theory :: Primes)

Pending Errata and Addenda

None.

Discussion

forum policy

No messages.

Interact