2023-02-riemann-zeta-properties.md

title	description	date
the riemann hypothesis	The Riemann hypothesis is considered one of the most important unsolved problems in pure mathematics, with wide-ranging applications across quantum physics, cryptography, number theory, and other fields. It is one of the [millenium prize problems][millenium-prize] by the [Clay Mathematics Institute][clay-institute]. At the heart of the hypothesis is the Riemann zeta function; ```math \begin{aligned} \zeta \;\colon \CC &\to \CC \\ s &\mapsto \sum_{n=1}^\infty \frac{1}{n^s} = \frac{1}{1^s} + \frac{1}{2^s} + \frac{1}{3^s} + \dotsi \\ \end{aligned} ``` What makes this function so interesting and consequential across scientific fields? [millenium-prize]: https://www.claymath.org/millennium-problems [clay-institute]: https://www.claymath.org/	2023-12-30 15:00:00 -0800

The complex field, denoted $\CC$, 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:

$$\begin{aligned} (a + ib) + (c + id) &= (a + c) + i(b + d) \\\ (a + ib) \times (c + id) &= (ac - bd) + i(ad + bc) \end{aligned}$$

Real Part Function

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

$$\begin{aligned} \Re \ \colon \CC &\to \RR \\\ a + ib &\mapsto a. \end{aligned}$$

Riemann Zeta Definition

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

$$\begin{aligned} \zeta(s) &= \sum_{n=1}^\infty \frac{1}{n^s} = \frac{1}{\Gamma(s)} \int\limits_0^\infty \frac{x^{s-1}}{e^x - 1} \,\d x \end{aligned}$$

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

$$\begin{aligned} \Gamma(s) &= \int\limits_0^\infty t^{s-1} e^{-t} \, \d t \end{aligned}$$

::alert

title: side quest

Show that $\displaystyle \sum_{n=1}^\infty \frac{1}{n^s}$ and $\displaystyle \frac{1}{\Gamma(s)} \int\limits_0^\infty \frac{x^{s-1}}{e^x - 1} ,\d x$ are indeed equivalent.

Here's some further context about gamma extensions that might be useful. ::

The Riemann Hypothesis

The zeta zeroes are what make the zeta function particularly interesting. $\zeta$ has trivial zeros² at $s = -2, -4, -6, \dotsc$. 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 $\Re(s) = \frac{1}{2}$.

::alert

type: warning title: relevance

Countless theories in fields as far apart as number theory, quantum mechanics, and cryptography assume that the Riemann hypothesis is true. ::

Relevance to Primes

Globality of Primes

In $1737$, Leonhard Euler proved that $\displaystyle \sum_{n=1}^\infty \frac{1}{n^s}$ is equivalent to $\displaystyle \prod_{p \text{ prime}} \frac{1}{1 - \frac{1}{p^s}}$, 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 $s = 1$;

$$\begin{aligned} \zeta(1) = \sum_{n=1}^\infty \frac{1}{n} &= \prod_{p \text{ prime}} \frac{1}{1 - \frac{1}{p}}, \end{aligned}$$

where $\displaystyle \sum_{n=1}^\infty \frac{1}{n}$ 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 $1$.

::alert

title: side quest

Can you think of an alternative proof of Euclid's theorem?

Euclid's Alternate Proof

Suppose there are only finitely many primes. Let $p$ be the largest prime. Take $\displaystyle N~=(2\times3\times5\times~\dotsm~\timesp)+~1$. Then $N-1$ is divisible by all primes less than $p$, so $N$ must not be divisible by any of those primes (since it would leave a remainder of $1$).

Therefore, $N$ is either prime or divisible by a prime greater than $p$, which contradicts the fact that we picked $p$ to be the largest possible such prime.

::

Locality of Primes

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:

$$\begin{aligned} \pi(x) &= \sum_{p \in \mathcal{P}_{\leq x}} 1 \end{aligned}$$

where $\mathcal{P}{\leq x}$ 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 $\RR{\geq 0}$ that starts at $0$ and increases by $1$ at each prime number.

Riemann developed a prime-power counting function $\displaystyle \Pi_0(x)=\frac{1}{2} \parens{ \sum_{p^n < x} \frac{1}{n} + \sum_{p^n \leq x} \frac{1}{n} }$. 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.

Footnotes

1. A pole of a function $f$ is a point in the domain of $f$ where the value of $f$ is undefined. ↩

2. A zero of a function $f$ is a value $z$ such that $f(z) = 0$. A trivial zero is a zero that is particularly uninteresting. ↩

3. Riemann was, in fact, Gauss's student. ↩