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
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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
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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.
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0@mixe$d$math: Thanks for the helpful a$d$ditio$n$. – 2011-08-24