Mangoldt summatory function

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:

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

where $\mathrm{\Lambda }$ 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 $\mathrm{\Lambda }(x)$ 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.