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:

$$ \psi(x) = \sum_{r \leq x} \Lambda(r) $$

where $\Lambda(x)$ 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:

$$ \pi(x) \sim \frac{x}{\ln(x)} $$

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

$$ \psi(x) \sim x $$

We can also define a ``smoothing function'' for the summatory function, defined as:

$$ \psi_1(x) = \int_0^x \psi(t) dt $$

and then the prime number theorem is also equivalent to:

$$ \psi_1(x) \sim \frac{1}{2} x^2 $$

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