Chebyshev functions

Summary

The Chebyshev functions $\vartheta(x)$ and $\psi(x)$ count the prime numbers up to $x$ with the weight $\log p$. The first sums $\log p$ over primes $p \le x$, and the second also counts the prime powers $p^k \le x$. They are easier to handle analytically than the prime-counting function $\pi(x)$, and the prime number theorem is equivalent to $\psi(x) \sim x$.