Definition:Second Chebyshev Function

Definition
The Second Chebyshev Function $\psi \left({x}\right)$ is defined as follows:


 * $\displaystyle \psi \left({x}\right) = \sum_{k \mathop \ge 1} \sum_{p^k \mathop \le x} \ln p$

where, for each $k$, the sum extends over all powers of prime numbers $p$ such that $p^k \le x$.

Equivalent definitions
The following are equivalent to the above definition:


 * $\displaystyle \psi \left({x}\right) = \sum_{1 \mathop \le n \mathop \le x} \Lambda \left({n}\right)$

where $\Lambda$ is the von Mangoldt function.

where the sum extends over all prime numbers $p$ such that $p \le x$, and $\left \lfloor {\cdot} \right \rfloor$ denotes the floor function.
 * $\displaystyle \psi \left({x}\right) = \sum_{p \mathop \le x} \left \lfloor {\log_p x} \right \rfloor \ln p$

Also See
Equivalence of Definitions of the Second Chebyshev Function