Exercise 1140Number theory › PrimesL1Added 3 Nov 2026

The prime counting function

#prime counting#pi function#estimates

Statement

Compute $\pi(30)$ where $\pi(x)$ is the number of primes $\le x$ .

Hint

Use the sieve of Eratosthenes.

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Primes up to 30. $2,3,5,7,11,13,17,19,23,29$ .

Count. $\pi(30)=10$ .

Comparison. The prime number theorem gives $\pi(x)\sim x/\ln x$ ; at $x=30$ , $30/\ln30\approx8.8$ , close to $10$ .

Conclusion. $\pi(30)=10$ .