Riemann zeta function

The Riemann zeta function or Euler–Riemann zeta function, $ζ (s)$ , is a mathematical function of a complex variable $s$ , and can be expressed as:

ζ (s) = \sum_{n = 1}^{\infty} \frac{1}{n^{s}} = \frac{1}{1^{s}} + \frac{1}{2^{s}} + \frac{1}{3^{s}} + \dots, if Re (s) > 1

For $Re (s) \leq 1$ is defined as the unique analytic continuation.

See this video for more info.

It is known that $- 2 k$ are zeroes of $ζ$ but there are others. Also it is known that any other zero must lie in

{s \in C : 0 < Re (s) < 1}

called the critical strip. The set ${s \in C : 0 < Re (s) = \frac{1}{2}}$ is called the critical line, and the Riemann hypothesis asserts that all the other zeroes lie in it. This has to do with prime numbers.

The zeroes of Riemann zeta function are important because they give the Fourier transform of a function that counts the distribution of primes.

This is also related to Mellin transform, but I don't understand yet...

How is this related to prime numbers? The imaginary part of the zeroes of $ζ$ are related to the distribution of primes. We can construct a kind of series with cosine and logarithms that gives rise to kind of wave whose peaks appear in the primes or prime powers. The zeroes of $ζ$ are related to the frequency of this wave. See this video.