Before we begin, you should know that I’m not actually going to present a proof of the Riemann Hypothesis.

This article is about a fictional object known as the field with one element, sometimes denoted Fᵤₙ. You can probably guess why: F is for field and “un” is the French word for 1.

When I first heard about this in grad school, I thought it was a joke. The object is “Fun” and it doesn’t really exist. How could people actually take this seriously?

But a lot of great mathematicians have done quite a bit of work on it: Jacques Tits, Alain Connes, Yuri Manin, and more.

A proper formulation could have major consequences for multiple branches of math, including computational complexity theory, noncommutative geometry, Arakelov geometry, and algebraic number theory (of course others too).

For the purpose of this article, we’ll focus on why it could possibly shed some light on the Riemann Hypothesis. So let’s get started.

What are fields?

Fields are one of the fundamental objects of mathematics. They are just an abstraction of what we already can do with the real numbers.

A field is a set with two operations (addition and multiplication) in which there is an additive identity, 0, and a multiplicative identity, 1, and every element (except 0) has an inverse for both operations.

That’s a lot of words, but it’s much simpler than it looks if you just think about the real numbers, ℝ, as the model for this.

Every real number has an additive inverse. For example, the additive inverse of 3 is just -3 because 3+(-3)=0.

Every non-zero real number has a multiplicative inverse. For example, the multiplicative inverse of 3 is (1/3). This is because 3*(1/3)=1.

For the same reason, ℚ, the set of rational numbers is also a field. So is ℂ, the set of complex numbers (imaginary numbers). If you’ve never seen this, try to work out things like the multiplicative inverse of 1+i to convince yourself.

Not all common numbers systems are fields. The set of integers, ℤ, do not form a field because, although there are addition and multiplication operations, 3 has no multiplicative inverse (1/3 is not an integer).

Not all fields are infinite. In fact, the integers with “mod 3” arithmetic, denoted ℤ/3, is a field with three elements {0,1,2}.

Mod 3 arithmetic is where you add and multiply as usual and then “loop around” like a clock: 0,1,2, 3=0, 4=1, 5=2, 6=0, …. (negatives work too: -1=2, etc).

We can check:

• 1+2=0. This shows that 1 is the additive inverse of 2 (and vice-versa).
• 2*2=4=1. This shows that 2 is its own multiplicative inverse.

Those were the only two difficult numbers to check, and so ℤ/3 is indeed a field.

Fields are quite well-understood at this point. Notice something weird happens in the finite field case. If you add 1 to itself 3 times you get the additive identity: 1+1+1=0. But if you do this in the infinite examples I showed, this never happens.

If you add 1 to itself a finite number of times and get 0, the number of times you need to do this is called the characteristic of the field. We’ll denote that with the letter p.

The characteristic of ℤ/3 is 3.

If adding 1 to itself never gives you 0 no matter how many times you do it, the characteristic is 0.

Important terminology: I’ll keep referring to positive characteristic in this article. This is just the standard way of referring to characteristic not equal to 0. In other words, p>0.

Most first courses on abstract algebra will prove an amazing fact: In the finite case, the characteristic is always a prime number! Moreover, the number of elements in a finite field is always a prime power: pⁿ (and p is the characteristic of such a field).

Conversely, for any prime power, there is an explicit construction for the field with that many elements. So, there are fields with 2 elements, 3 elements, 4 elements, 5 elements, 7, 8, 9, etc. There are no fields with 6 elements or 10 elements.

For completeness sake, not all infinite fields have characteristic 0. It’s easy to get the wrong impression from this limited explanation.

This brings us to a sticking point: There is no field with one element! (Namely, 1 is not a prime power).

Before getting more into that, let’s take a detour to see why one might hope there is such a thing.

The Riemann Hypothesis over finite fields

This is where things get really cool.

It turns out that “doing geometry” over positive characteristic fields is often easier than in characteristic 0 (but also harder in other senses). I won’t get into why, since that’s an entire graduate course of subtlety.

So it is sometimes a great tool to take something you want to prove over ℂ and reduce to positive characteristic, prove it there, and then try to lift it back to characteristic 0 somehow (this was actually a major point of my thesis).

It’s a bit complicated to define what geometry means in positive characteristic, but we can rely on a fairly accurate analogy. Geometry over ℝ or ℂ just means studying the shapes formed by the zero sets of polynomials.

So, if you’re over ℝ, have 2 variables, and a polynomial p(x,y)=y-x², the geometry it makes is the parabola you get when setting that equal to zero:

y-x²=0
Or more familiar to people: y=x²

I’ve talked about some of the weirdness of doing this over ℂ in other articles (The Hodge Conjecture and Falting’s Theorem and the Mordell Conjecture).

When you do this over ℚ, you get an interesting interplay between the topology and the number of integer solutions (Fermat’s Last Theorem and so on).

We can just do the same thing over finite fields. The geometry of p(x,y)=y-x² over ℤ/3 comes from plugging in and checking the zero set. It’s harder to visualize as “geometry”, but actually easier to work with since it’s finite.

In fact, we can just check {(0,0), (1,1), (2,1)} are the only three points. I’m glossing over some important details (like doing this projectively and making sure it’s nonsingular), but this way of thinking is good enough for getting the gist of things.

Zeta Functions

Suppose we start with a finite field with p elements, say F, and a “curve,” C, over that field (the zero set of a polynomial for simplicity).

We can count the number of points C has, N(1).

Then we can look at the same equation over the field with p² elements and call this N(2) and so on.

So N is a function. N(k) is just the number of points of C over the field with pᵏ elements.

The next part will look complicated, but it will simplify greatly.

Consider the function G(t)=N(1)t+N(2)t²/2+N(3)t³/3+…

The local Zeta function of C is defined as the exponential of that:

Z(C, t)=exp(G(t)).

This might look crazy, but we can work out one example very easily to see that the definition is constructed to make things cancel out and simplify.

If we start with the polynomial p(x)=x, then the only solution to x=0 over any field is just x=0.

There is only one point no matter how many fields we check. Therefore, N(k)=1 for all k.

Let’s plug that in:

G(t)=t+t²/2+t³/3+…=-ln(1–t) for well-defined t, by basic Calculus (check the Taylor series).

Thus, Z(C, t)=exp(-ln(1-t))=1/(1-t).

Wow! See, it’s not bad at all.

The Weil Conjectures are a bunch of conjectures about Z(X, t) for any X (not just curves or points but higher-dimensional spaces, too) proposed by André Weil in 1949. They have been one of the main driving forces in research in algebraic/arithmetic geometry ever since.

Dwork, Grothendieck, and Deligne ended up proving them over several decades and many modern alternate proofs have been found.

The key takeaway is that one of the Weil Conjectures is the “Riemann Hypothesis” for these zeta functions. The general one is extremely technical, but Weil himself proved the Riemann Hypothesis for curves over finite fields.

The proof is relatively easy with the appropriate geometric machinery (for example, it’s left as an exercise in Hartshorne’s book Algebraic Geometry).

The Field With One Element

Okay. We’re finally ready to talk about the field with one element.

Remember, it doesn’t exist.

But the idea is to construct something that would let us do a type of generalized geometry.

Think about ℤ. It has the property that “reducing mod p,” for any prime p gives us the field with p elements (this is basically how we defined the field with p elements earlier).

This fact can be restated geometrically. There is a geometric space, X=Spec(ℤ), so that reducing mod p gives us exactly 1 point over the field with p elements, Fₚ, for each prime number p.

We worked out the local zeta function for a single point already! It’s just: 1/(1-t).

But we’re reducing these mod p to get a “local” zeta function. When you bundle these together to get the “global” zeta function of X, the appropriate way of doing this is by multiplying and keeping track of the prime (t→p⁻ˢ), we get:

the Riemann zeta function (from Riemann Hypothesis fame) in product form!

This probably looks ad hoc and random, but there are better-streamlined definitions that make this more natural and motivated.

This is what we’ve been looking for.

What if there were something called Fᵤₙ that acted like a finite field in such a way that we could treat X=Spec(ℤ) as a curve over it and then use Weil’s techniques for curves over finite fields to prove the Riemann Hypothesis?

People have done some pretty amazing work in this direction. Maybe one day we’ll have a Fᵤₙ proof of the Riemann Hypothesis.