A number theoretic function used in the study of prime numbers; specifically it was used in the proof of the prime number theorem.

It is defined thus:

where $\Lambda$ is the Mangoldt function.

The Mangoldt summatory function is valid for all positive real x.

Note that we do not have to worry that the inequality above is ambiguous, because $\Lambda(x)$ is only non-zero for natural $x$. So no matter whether we take it to mean r is real, integer or natural, the result is the same because we just get a lot of zeros added to our answer.

The prime number theorem, which states:

where $\pi(x)$ is the prime counting function, is equivalent to the statement that:

We can also define a “smoothing function” for the summatory function, defined as:

and then the prime number theorem is also equivalent to:

which turns out to be easier to work with than the original form.