the riemann hypothesis

The complex field, denoted $\mathbb{C}$, is the set of all the complex numbers of the form $\sigma +it$, where $\sigma$ and $t$ are real numbers. Addition and multiplication over the field are defined as follows:

Over $\mathbb{C}$, we define the real function $\mathrm{Re}(s)$ as follows:

For $\mathrm{Re}(s)>1$, $\zeta$ is defined as an absolutely convergent infinite series that maps values from $\mathbb{C}$ unto itself, with the exception of $s=1$ where the sum diverges and $\zeta$ has a pole¹.

where $\Gamma$ is the Gamma function, a particularly useful analytic continuation of factorials to non-integral points.

The zeta zeroes are what make the zeta function particularly interesting. $\zeta$ has trivial zeros² at $s=-2,-4,-6,\dots$. Besides these, all the other zeroes, so far, have been found to have real part $\frac{1}{2}$.

The Riemann hypothesis, which remains unproven, asserts that indeed ALL the non-trivial zeroes have $\mathrm{Re}(s)=\frac{1}{2}$.

In $1737$, Leonhard Euler proved that $\sum _{n=1}^{\infty }\frac{1}{n^s}$ is equivalent to $\prod _{p\text{ prime}}\frac{1}{1-\frac{1}{p^s}}$, which directly relates the zeta function to the prime numbers. For example, this result can be used to construct a direct proof of Euclid's theorem, which posits that there are infinitely many primes, by equating the two equations at $s=1$;

where $\sum _{n=1}^{\infty }\frac{1}{n}$ is the harmonic series that diverges to infinity. Thus, the product must also diverge to infinity, which implies an infinite number of terms terms in the product that are greater than $1$.

The above equation gives an estimate of the global distribution of primes and them being unbounded. However, a much stronger result which directly relates to the Reimann hypothesis itself estimates the locality of primes.

Gauss³ posited the prime-counting function $\pi (x)$ as follows:

where ${\mathcal{P}}_{\le x}$ is the set of all prime numbers less than $x$. This $\pi$ is more commonly referred to as the prime-counting function. That is, it is a step function over ${\mathbb{R}}_{\ge 0}$ that starts at $0$ and increases by $1$ at each prime number.

Riemann developed a prime-power counting function ${\Pi }_0(x)=\frac{1}{2}\left(\sum _{p^n<x}\frac{1}{n}+\sum _{p^n\le x}\frac{1}{n}\right)$. Riemann then showed that the zeta zeroes can be used to very accurately approximate the locality of primes. Here's a nice discussion of the consequences.