# 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:

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

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:

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

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.

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