Equivalence of Definitions of Second Chebyshev Function

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Theorem

The following definitions of the concept of Second Chebyshev Function are equivalent:

$(1): \quad \displaystyle \psi \left({x}\right) = \sum_{p^k \mathop \le x} \ln p$
$(2): \quad \displaystyle \psi \left({x}\right) = \sum_{1 \mathop \le n \mathop \le x} \Lambda \left({n}\right)$
$(3): \quad \displaystyle \psi \left({x}\right) = \sum_{p \mathop \le x} \left \lfloor {\log_p x} \right \rfloor \ln p$

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:

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

follows directly from the definition of the von Mangoldt function.

$\Box$


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:

\(\displaystyle \sum_{n \mathop = 1}^N \Lambda \left({n}\right)\) \(=\) \(\displaystyle \Lambda \left({1}\right) + \Lambda \left({2}\right) + \cdots + \Lambda \left({N}\right)\)
\(\displaystyle \) \(=\) \(\displaystyle 0 + \ln \left({2}\right) + \ln \left({3}\right) + \ln \left({2}\right) + \ln \left({5}\right) + 0 + \ln \left({7}\right) + \ln \left({2}\right) + \ln \left({3}\right) + 0 + \cdots\)

Notice this sum will have:

as many $\ln \left({2}\right)$ terms as there are powers of $2$ less than or equal to $N$,
as many $\ln \left({3}\right)$ 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:

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

$\blacksquare$