The book I am working through uses the bound
$\pi(x)>\frac{x}{ \log x}$
without proof.
Is it possible to prove this in a simple way using Sieve methods?
2026-03-25 20:41:39.1774471299
Lower bound on $\pi(x)$
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You might look into Chebyshev's elementary arguments to show that $$ \frac{\pi(x)\log x}{x} > 0.9,$$ which is not as good but which is rather easy to produce. I'm not sure what exactly you are looking for, but this may or may not suffice for your purposes.
To actually show that $\pi(x) > \frac{x}{\log x}$ is significantly more challenging.