Chebyshev functions

Remarks

Example: $\vartheta(10) = \log 2 + \log 3 + \log 5 + \log 7 = \log 210$. The prime powers up to $10$ are $2, 4, 8, 3, 9, 5, 7$, so $\psi(10) = \log(2^3 \cdot 3^2 \cdot 5 \cdot 7) = \log 2520$. In general $\psi(x) = \log \operatorname{lcm}(1, 2, \dots, \floor{x})$ for $x \ge 1$. Non-example: $\sum_{n \le 10} \log n = \log 10!$ is not $\psi(10)$, since it also has terms such as $\log 6$ for $n$ that are not prime powers. The two functions are related by $\psi(x) = \sum_{k \ge 1} \vartheta(x^{1/k})$, and $\psi(x) - \vartheta(x) = O(\sqrt{x} \log x)$. Both are $0$ for $x < 2$.