Definition:Second Chebyshev Function

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


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

where 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:


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

where $$\Lambda$$ is the von Mangoldt function.

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

Also See
Equivalence of Definitions of the Second Chebyshev Function