Second Chebyshev function

This article is about a function defined on positive reals (and in particular, natural numbers) obtained as the summatory function of an arithmetic function, namely von Mangoldt function.
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Definition

Let $x$ be a positive real number. The second Chebyshev function of $x$, denoted $\psi (x)$, is defined as the following sum:

$\psi (x)={\sum }_{n\le x}\mathrm{\Lambda }(n)$.

Here, $\mathrm{\Lambda }$ is the von Mangoldt function.

This summation is taken over all the natural numbers less than or equal to $x$; however, a positive contribution comes only from prime powers, and the contribution of a prime power $p^k$ is $\log p$.

Relation with other counting functions

• Prime-counting function
• First Chebyshev function