Equivalence of Definitions of Second Chebyshev Function

Theorem
These definitions of the second Chebyshev function are equivalent:


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


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


 * $\displaystyle \psi \left({x}\right) = \sum_{p \le x} \left \lfloor {\log_p x} \right \rfloor \ln p$

where:
 * $p$ is a prime number
 * $\Lambda \left({n}\right)$ is the von Mangoldt function
 * $\left \lfloor {\ldots} \right \rfloor$ denotes the floor function.

Proof

 * The equivalence:
 * $\displaystyle \sum_{p^k \le x} \ln p \equiv \sum_{1 \le n \le x} \Lambda \left({n}\right)$

follows directly from the definition of the von Mangoldt function.


 * Let $N = \left \lfloor {x} \right \rfloor$.

It can be seen directly that all the above summations are exactly the same whether performed on $N$ or $x$.

Hence we need only to prove the equivalence for integral arguments.

First we expand the von Mangoldt function:

Notice this sum will have:
 * as many $\ln(2)$ terms as there are powers of $2$ less than or equal to $N$,
 * as many $\log(3)$ terms as there are powers of $3$ less than or equal to $N$

and in general, if $p$ is a prime less than $N$, $\ln p$ will occur in this sum $\left \lfloor {\log_p N} \right \rfloor$ times.

Hence:
 * $\displaystyle \sum_{1 \le n \le x} \Lambda \left({n}\right) \equiv \sum_{p \le x} \left \lfloor {\log_p x} \right \rfloor \ln p$