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 functions

• Prime-counting function
• First Chebyshev function: This simply adds the logarithms of all the primes up to the point.

Exponential

The exponential of the second Chebyshev function gives the lcm of all numbers so far. In other words:

$e^{\psi (x)}=\mathrm{l}\mathrm{c}\mathrm{m}\{1,2,\dots ,[x]\}$

where $[x]$ denotes the greatest integer function of $x$, i.e., the largest integer less than or equal to $x$.