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.

Important property: Euler product for $ℜ (s) > 1$ . Idea:

ζ (s) = 1 + \frac{1}{2^{s}} + \frac{1}{3^{s}} + \frac{1}{4^{s}} + \frac{1}{5^{s}} + \frac{1}{6^{s}} + \frac{1}{7^{s}} + \frac{1}{8^{s}} + \dots

\frac{1}{2^{s}} ζ (s) = \frac{1}{2^{s}} + \frac{1}{4^{s}} + \frac{1}{6^{s}} + \frac{1}{8^{s}} + \frac{1}{10^{s}} + \frac{1}{12^{s}} + \frac{1}{14^{s}} + \frac{1}{16^{s}} + \dots

ζ (s) - \frac{1}{2^{s}} ζ (s) = 1 + \frac{1}{3^{s}} + \frac{1}{5^{s}} + \frac{1}{7^{s}} + \frac{1}{9^{s}} + \frac{1}{11^{s}} + \frac{1}{13^{s}} + \dots

(1 - \frac{1}{2^{s}}) ζ (s) = 1 + \frac{1}{3^{s}} + \frac{1}{5^{s}} + \frac{1}{7^{s}} + \frac{1}{9^{s}} + \frac{1}{11^{s}} + \frac{1}{13^{s}} + \dots

Same series without the terms multiple of two. And analogously:

\frac{1}{3^{s}} (1 - \frac{1}{2^{s}}) ζ (s) = \frac{1}{3^{s}} + \frac{1}{9^{s}} + \frac{1}{15^{s}} + \frac{1}{21^{s}} + \frac{1}{27^{s}} + \frac{1}{33^{s}} + \frac{1}{39^{s}} + \dots

(1 - \frac{1}{2^{s}}) ζ (s) - \frac{1}{3^{s}} (1 - \frac{1}{2^{s}}) ζ (s) = (1 - \frac{1}{2^{s}}) (1 - \frac{1}{3^{s}}) ζ (s)

= 1 + \frac{1}{5^{s}} + \frac{1}{7^{s}} + \frac{1}{11^{s}} + \frac{1}{13^{s}} + \frac{1}{17^{s}} + \frac{1}{19^{s}} + \frac{1}{23^{s}} + \frac{1}{25^{s}} + \dots

and we have removed the multiple of three. A kind of Eratostenes sieve.

If we continue, at the end we obtain:

\prod_{p prime} (1 - \frac{1}{p^{s}}) \cdot ζ (s) = 1,

or, also:

ζ (s) = \prod_{p prime} \frac{1}{1 - p^{- s}} .

Another motivation

Let's think about the number of primes up to 100. By inclusion-exclusion (like the formula for the measure of $A \cup B \cup C$ ):

Number of primes = 100 - (\frac{100}{2} + \frac{100}{3} + \frac{100}{5} + \frac{100}{7})

+ (\frac{100}{2 \cdot 3} + \frac{100}{2 \cdot 5} + \frac{100}{2 \cdot 7} + \frac{100}{3 \cdot 5} + \frac{100}{3 \cdot 7} + \frac{100}{5 \cdot 7})

- (\frac{100}{2 \cdot 3 \cdot 5} + \frac{100}{2 \cdot 3 \cdot 7} + \frac{100}{2 \cdot 5 \cdot 7} + \frac{100}{3 \cdot 5 \cdot 7}) + (\frac{100}{2 \cdot 3 \cdot 5 \cdot 7})

And so,

Number of primes \approx 100 \cdot (1 - \frac{1}{2}) \cdot (1 - \frac{1}{3}) \cdot (1 - \frac{1}{5}) \cdot (1 - \frac{1}{7})

What happens if we want to understand the density of primes not just up to 100, but across the entire infinite set of natural numbers?

We must extend our sieve infinitely, applying it to every prime number that exists. The "survival probability" of a number being prime becomes an infinite product over all primes:

\prod_{p (prime)} (1 - \frac{1}{p})

This product equals zero (since the density of primes thins out to zero), but the structure of this equation caught the attention of the 18th-century mathematician Leonhard Euler. What if we invert this factors?
Using

\frac{1}{1 - \frac{1}{p}} = 1 + \frac{1}{p} + \frac{1}{p^{2}} + \frac{1}{p^{3}} + \dots

we obtain

\frac{1}{\prod_{p (prime)} (1 - \frac{1}{p})} = \sum_{n = 1}^{\infty} \frac{1}{n}

There was one problem: the harmonic series ( $\sum 1 / n$ ) diverges to infinity. To make this equation mathematically useful and convergent, Euler introduced an exponent, $s$ , making the terms shrink fast enough:

\sum_{n = 1}^{\infty} \frac{1}{n^{s}} = \prod_{p (prime)} (\frac{1}{1 - p^{- s}})

This left-hand side is the definition of the Riemann Zeta function, $ζ (s)$ .

Connection with the prime counting function

To connect with the prime counting function and in particular with the Chebyshev function, observe that

- \frac{ζ^{'} (s)}{ζ (s)} = \sum_{n} \frac{Λ (n)}{n^{s}},

where

Λ (n) = {\begin{cases} \log p & if n = p^{k} for some prime p and k \geq 1, \\ 0 & otherwise . \end{cases}

So this logarithmic derivative of $ζ$ contain info about the "accumulation of primes until the end". See prime counting function#Dream of a Formula.

About the zeroes

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 a kind of Fourier transform of a function that counts the distribution of primes: the prime counting function.

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.