Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Does there exist a function $f$ that is a lower bound of the prime number function $\pi$ with $f \sim \pi$?

share|improve this question
2  
Just to be a little puckish, I will note that $$f(x)=\pi(x)-1$$ satisfies your requirements :) –  Zev Chonoles Aug 23 '11 at 18:10
    
Yep, thanks. My wording should be more precise... –  testito Aug 23 '11 at 18:36
add comment

2 Answers 2

This segment of the Wikipedia article mentions a few such bounds. In all cases, the bounds hold from a certain explicit $N$ on. They can be easily be made unconditional without changing the asymptotic behaviour by suitably redefining the functions for $x \lt N$.

For instance, the article mentions the bounds $$\frac{x}{\ln x+2}<\pi(x)<\frac{x}{\ln x-4},$$ valid for $x \ge 55$, as well as much stronger bounds by Pierre Dusart. Since $\pi(x)$ is asymptotically $x/\ln x$, the two bounds above have the right asymptotic properties. Tweaking the lower bound so that it is valid below $55$ is easy. The crudest method is to use the function which is $-1$ for $x \lt 55$, and $x/(\ln x +2)$ for $x \ge 55$. Asymptotic behaviour is unaffected.

share|improve this answer
1  
See also math.stackexchange.com/questions/54312/…. –  lhf Aug 23 '11 at 18:20
1  
I wanted to add the article of Dussart. –  mixedmath Aug 24 '11 at 11:05
    
@mixedmath: Thanks for the helpful addition. –  André Nicolas Aug 24 '11 at 11:10
add comment
up vote 1 down vote accepted

I found an more 'elegant' lower bound:

$$ \frac{n}{\log\,n} - 2 \leq \pi(n), \; n \geq 2 $$ Primality Testing in Polynomial Time

share|improve this answer
    
There is a proof on page 46: springerlink.com/content/p9rft7x5nkbwjkq6 –  testito Sep 24 '11 at 1:41
    
Sorry, was thinking of $\text{li}(x)$, that bounces back and forth from below $\pi(x)$ to above. –  André Nicolas Sep 24 '11 at 1:49
    
Although, I am not sure what is meant by $\log$ in the paper. The author distinguishes between $\ln$ (natural logarithm) and $\log$ (base 2 or base 10, maybe?)... –  testito Sep 24 '11 at 1:57
    
On second or third thought, I am again very surprised at the result, and doubtful. Maybe there is confusion with estimates of $p_n$. I do not trust Wolfram Mathworld assertions, they are often wrong. –  André Nicolas Sep 24 '11 at 6:18
    
mmh, I don't think so, Martin Dietzfelbinger seems like a reliable source. –  testito Sep 24 '11 at 12:20
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.