## Does $π(x) \ge \log \log x$ hold for $2 \le x \le e^{e^3} The book Theory of Numbers by G H Hardy, et al. proves $$π(x) \ge \log \log x$$ for $$x > e^{e^3}$$. Is there a way to prove that it also holds for $$2 \le x \le e^{e^3}$$ otherwise (at worst) any valuable source to check this numerically? ## Does finding a cycle with$\log n$length in$\text{P}\$ ?

Let $$G$$ be an arbitrary graph with $$n$$ vertices and we want to find a cycle with $$\log n$$ length. Is there exists a known polynomial algorithm for this problem?