Equivalence of Definitions of the 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:

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

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:

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

$\blacksquare$