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:

(1)

\begin{align} \quad \psi (n) = \sum_{n \leq x} \Lambda (n) \end{align}