Does there exist a function $f$ that is a lower bound of the prime number function $\pi$ with $f \sim \pi$?
lower bound for the prime number function
-
5Just to be a little puckish, I will note that $$f(x)=\pi(x)-1$$ satisfies your requirements :) – 2011-08-23
-
0Yep, thanks. My wording should be more precise... – 2011-08-23
2 Answers
I found an more 'elegant' lower bound:
$$ \frac{n}{\log\,n} - 2 \leq \pi(n), \; n \geq 2 $$ Primality Testing in Polynomial Time
-
0There is a proof on page 46: http://www.springerlink.com/content/p9rft7x5nkbwjkq6/ – 2011-09-24
-
0Sorry, was thinking of $\text{li}(x)$, that bounces back and forth from below $\pi(x)$ to above. – 2011-09-24
-
0Although, 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?)... – 2011-09-24
-
0On 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. – 2011-09-24
-
0mmh, I don't think so, Martin Dietzfelbinger seems like a reliable source. – 2011-09-24
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.
-
1See also http://math.stackexchange.com/questions/54312/non-trivial-upper-bound-for-the-number-of-primes-less-or-equal-to-n/54325#54325. – 2011-08-23
-
1I wanted to add the [article](http://arxiv.org/PS_cache/arxiv/pdf/1002/1002.0442v1.pdf) of Dussart. – 2011-08-24
-
0@mixedmath: Thanks for the helpful addition. – 2011-08-24