Warning: naïve question ahead!
Littlewood proved in 1914 that there exists a number $n\in\mathbb{N}$ (called Skewes' number) such that:
$$ \pi(n) > \operatorname{li}(n). $$
It is conjectured that $n$ is a huge number, recent analysis suggests $n\approx e^{727.951}$. Since then, researchers have worked to find lower and upper bounds for $n$. Currently it is held that:
$$ 10^{19}<n<e^{727.951}. $$
Why is it that researchers are trying to improve these bounds instead of finding an exact value? Since $\pi(n) \sim \frac{n}{\ln(n)}$ for very large numbers (i.e. > $10^{19}$), we have:
$$ \frac{n}{\ln(n)} > \int_0^n \frac{\mathrm dt}{\ln t}. $$
And with sufficiently big computing power this should not be too hard to solve. What am I missing?
$\endgroup$ 52 Answers
$\begingroup$You have several problems with this question. One is that replacing functions with values that are asymptotically equal is not justified. Consider $f(x)=x^2, g(x)=x^2+\frac 12, h(x)=x^2+\cos x$. They are all asymptotically $x^2$ but comparing them depends on the non-leading terms. Second, $\operatorname{li}(n)$ is a much better approximation to $\pi(n)$ than $\frac n{\ln (n)}$. Skewes' number depends on the comparison between $\pi(n)$ and $\operatorname{li}(n)$. $\frac n{\log n}$ is not involved. Your final inequality is false for $n \gt 3.847$ Finally, you seem to expect that $10^{19}$ or Skewes' number is large enough for asymptotics to have taken hold. It would not be hard to construct an example where this fails. We could ask for what $n$ does $10^{-6}\log (\log(n))$ exceed $1$. It obviously does eventually, but for much larger values than we are talking about here.
$\endgroup$ 1 $\begingroup$A practical answer is this. The current best estimate of the first Skewes number is ${SK}_{1} = 1.397162914 \times 10^{316}$. Now computing on Mathematica the logarithmic integral at this value takes a fraction of a second. However the efforts in computing the exact value of the prime counting function at the value of $10^27$ results in a paper (2015 by David Baugh and Kim Walisch) Walisch, Kim (September 6, 2015). "New confirmed pi(10^27) prime counting function record". Mersenne Forum. So at this point no one can compute the exact value of the prime counting function near the first Skewes number within a reasonable amount of time. This is why say lets sprinkle random numbers near Skewes first number, compute the values and see if we 'hit" a point where the prime count is larger than the logarithmic integral has not occurred.
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