Riemann Hypothesis

December 30, 2023

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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 by the Clay Mathematics Institute.

At the heart of the hypothesis is the Riemann zeta function;

\begin{aligned} \zeta \;\colon \C &\to \C \\ 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?

The complex field, denoted $\C$ , 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:

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

\begin{aligned} \Re \ \colon \C &\to \R \\ 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 $\C$ 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}

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 $\Re(s) = \frac{1}{2}$ .

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 $1$ .

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 $\displaystyle N~=~(2~\times~3~\times~5~\times~\dotsm~\times~p)~+~1$ . 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 $\R_{\geq 0}$ 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.

A pole of a function $f$ is a point in the domain of $f$ where the value of $f$ is undefined. ↩
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. ↩
Riemann was, in fact, Gauss's student. ↩

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