The Chebyshev Functions
 Definition: The First Chebyshev Function or Theta Function is the function $\theta : X \to \mathbb{R}$ defined for all $x \in \mathbb{R}$ by $\displaystyle{\theta(x) = \sum_{p \leq x} \ln p}$.
 Definition: The Second Chebyshev Function or Psi Function is the function $\psi : X \to \mathbb{R}$ defined for all $x \in \mathbb{R}$ by $\displaystyle{\psi(x) = \sum_{p^k \leq x} \ln p}$.
Note that if $p^k \leq x$ then $\Lambda (p^k) = \ln p$ and otherwise, $\Lambda (n) = 0$. Therefore the psi function can be expressed alternatively by the formula: