\begin{definition}[Definition:Second Chebyshev Function/Definition 1]
The '''second Chebyshev Function''' $\psi: \R \to \R$ is defined as:
:$\ds \forall x \in \R: \map \psi x := \sum_{k \mathop \ge 1} \sum_{p^k \mathop \le x} \ln p$
where, for each $k$, the summation extends over all powers of prime numbers $p$ such that $p^k \le x$.
\end{definition}