This blog post consists of three parts. The first of them contains a somewhat nontechnical description of Riemann hypothesis. In the second one we discuss what the “correct” analogue of Riemann hypothesis is for polynomials over a finite field. Finally, in the last section, we prove the Riemann hypothesis for polynomials.

Riemann zeta function and Riemann hypothesis

Consider the following infinite series, which depends on the number \(s\), which, at first, we take to be real:

\(\displaystyle\sum_{n=1}^\infty\frac{1}{n^s}=\frac{1}{1^s}+\frac{1}{2^s}+\frac{1}{3^s}+\dots\).

After writing this down, the first question which should be asked whether this makes sense, and the answer (rather clearly) depends on \(s\). It’s easy to see, thanks to Cauchy condensation test, that this series converges precisely for \(s>1\). We denote the sum of this series by \(\zeta(s)\) and call it the Riemann zeta function.

Now let’s remove the constraint that \(s\) should be real. It’s not obvious what \(n^s\) should mean when \(s\) is a complex number, but there is a way to define it consistent with exponentiation of real numbers and satisfies desired properties. Writing \(s=x+iy,x,y\in\mathbb R\), it is then true that \(\left|\frac{1}{n^s}\right|=\frac{1}{n^x}\). Therefore, for \(x>1\), this series converges absolutely. Thus we can define \(\zeta(s)\) for all complex numbers \(s\) with real part above \(1\).

Just like it is possible to extend exponentiation, initially defined only for real numbers, to all complex numbers, it is reasonable to ask whether we can do the same for \(\zeta(s)\) – is there a “reasonable” way to extend it to whole complex plane? Unfortunately, \(\zeta(s)\) tends to infinity as \(s\) tends to \(1\), so we shouldn’t expect there to be a “nice” way to define \(\zeta(s)\). However, there is a way to define \(\zeta(s)\) for all complex numbers other than \(1\). This extension is uniquely characterized by being differentiable at every point where it’s defined, which, for complex numbers, turns out to be a very stringent property. This unique extension is called the analytic continuation of \(\zeta(s)\).

So defined \(\zeta(s)\) is a function which is very well-understood for all \(s\) with real part greater than \(1\) or smaller than \(0\). For example, it is true (and not hard to show with proper tools) that in these parts of the complex plane the only solutions of \(\zeta(s)=0\) are negative even integers. These zeros of \(\zeta(s)\) are called trivial, not because it’s obvious that they are there, but because it is very easy to work with them.

Behaviour for \(s\) with real part between \(0\) and \(1\) (the region called the critical strip) is much more complicated. It is known that there are infinitely many more zeros here, and all of these are called nontrivial. The first 10 trillion (!) nontrivial zeros are known to all have real part equal to \(\frac{1}{2}\) – they lie right in the middle of the critical strip, on the critical line. This overwhelming computational evidence suggests the following conjecture:

Riemann hypothesis: All nontrivial zeros of Riemann zeta function lie on the critical line.

Of course, this conjecture might seem like some random statement about some random function, so why all the fuss about it? It turns out that the Riemann zeta function is very closely connected to the multiplicative structure of natural numbers and its zeros “control”, in a very specific meaning of this word, the distribution of prime numbers. Even trying to explain why Riemann zeta function should have anything to do with the distribution requires a large amount of complex analysis. Just to illustrate the importance of this theorem, I will leave here an equivalent statement of Riemann hypothesis.

Riemann hypothesis: Let \(\pi(x)\) denote the number of primes smaller than \(x\), and let \(\displaystyle\mathrm{Li}(x)=\int_2^x\frac{dt}{\ln t}\) (this function is approximately \(\frac{x}{\ln x}\)). Then, \(\pi(x)\) is very well-approximated by \(\mathrm{Li}(x)\). More precisely, for \(x\geq 2657\),

\(\displaystyle|\pi(x)-\mathrm{Li}(x)|<\frac{1}{8\pi}\sqrt{x}\ln x\)

Zeta function and Riemann hypothesis for polynomials

The idea we want to pursuit now is replacing the natural numbers with polynomials. It has turned out to be quite a fruitful idea in number theory, as polynomials over a finite field appear to have a similar structure as the integers. We will focus our attention on a single finite field, say with \(q\) elements. We will denote it by \(\mathbb F_q\).

First thing to realize is that \(\mathbb F_q[x]\) is more analoguous to \(\mathbb Z\) than to \(\mathbb N\). Hence we would like to pick out some elements of \(F_q[x]\) to represent the “natural numbers”. The correct way of doing this is with the help of units – the elements which have multiplicative inverses. In case of \(\mathbb Z\), the units are precisely \(\pm 1\) (note that, for example, \(\frac{1}{2}\not\in\mathbb Z\), so \(2\) is not a unit). We call a pair of elements which differ by a unit factor associates. So now we would like to somehow choose one element of associates pair \(\{n,-n\}\) for nonzero integers \(n\). There are two ways to proceed now. First of all is to make use of the fact that integers posses an ordering, so that we can speak of positive elements, and from each pair we choose a positive element.

In the polynomials over a finite field the units are precisely the nonzero constant polynomials, and there are \(q-1\) of them. So now the associate classes have \(q-1\) elements each. There happens to be quite a natural choice of an element from each class – the monic polynomial, i.e. the polynomial with the coefficient of the highest power of \(x\) equal to \(1\). The monic polynomials are closed under multiplication, though not under addition, so they don’t share all properties of \(\mathbb N\), but we can’t do any better.

But even at this point we realize that polynomials over finite fields live in a world very different from the world of real numbers (they are somewhat “incompatible”, for example, adding an element to itself a number of times gives us zero in \(\mathbb F_q[x]\)), so it’s difficult to imagine a way to take such a polynomial to a real or, better yet, complex power. We will return to this problem in a minute.

I’ve promised the second way to “choose” an element out of a group of associates. Here comes the best part – we don’t have to choose at all! Instead, we note that in each pair of associate integers both elements have the same absolute value, so instead of summing over natural numbers per se, we can sum over all these pairs and taking the absolute values of their elements. In order to apply this to polynomials, we have to make up the notion of absolute value which agrees over all associates. A first good guess is the degree \(\deg f\), but it lacks the multiplicative property which we would desire (instead, \(\deg fg=\deg f+\deg g\)). A better idea is to take \(q^{\deg f}\), so called norm of the polynomial. This is now a multiplicative function, and it’s a natural choice for one more reason – this is the number of congrunce classes modulo \(f\), which happens to agree with with the fact that \(|n|\) is the number of congruence classes modulo \(n\neq 0\). This idea can now be generalized in many directions, but we don’t pursuit that here.

Our problem of having to exponentiate polynomials solves itself with the second approach – instead of exponentiating a polynomial itself, we exponentiate its norm, which is a natural number. We can now (finally!) define the polynomial zeta function (for simplicity, we index the sum with monic polynomials instead of associate classes as discussed before):

\(\displaystyle\zeta_q(s)=\sum_{f\text{ monic}}\frac{1}{(q^{\deg f})^s}\)

This series needn’t be convergent for all values of \(s\), so we have to hope that it can be analytically continued to all or almost all complex numbers \(s\) like it is possible with \(\zeta(s)\). As with \(\zeta(s)\), we know that this extension, if it exists, will be unique, so (although technically we don’t know if the statement makes real sense) we can state the conjecture:

Polynomial Riemann hypothesis: All nontrivial (i.e. lying in the critical strip) zeros of \(\zeta_q(s)\) lie on the critical line.

Proof of polynomial Riemann hypothesis

There is one more technical detail regarding the definition of \(\zeta_q(s)\) – we have not specified the order in which the terms are being summed, and for infinite series the order of terms might make the result vary. To deal with this, we group the terms coming out of polynomials of the same norm. To be precise, let \(a_n\) be the number of polynomials having the norm \(n\). Then we can precisely define the zeta function to be

\(\displaystyle\zeta_q(s)=\sum_{n=1}^\infty\frac{a_n}{n^s}\)

Of course \(a_n=0\) for \(n\) not a power of \(q\) (because of how the norm is defined), and \(a_{q^d}\) is the number of monic polynomials of degree \(d\). To see what this is equal to, write a generic monic polynomial of degree \(d\) as \(x^d+c_{d-1}x^{d-1}+\dots+c_1x+c_0\). We have precisely \(q\) choices for each of \(d\) coefficients \(c_i\), so we see that there are \(q^d\) such polynomials, that is, \(a_{q^d}=q^d\). Therefore we can write

\(\displaystyle\zeta_q(s)=\sum_{d=0}^\infty\frac{q^d}{(q^d)^s}=\sum_{d=0}^\infty (q^{1-s})^d\)

This is a geometric series, and we know that it converges precisely when \(|q^{1-s}|<1\). Writing \(s=x+iy,x,y\in\mathbb R\) as on the beginning, \(|q^{1-s}|=q^{1-x}\), which is less than \(1\) iff \(x>1\). So for \(s\) with real part above \(1\) this series converges, and from well-known formula for the sum of a geometric series, we have

\(\displaystyle\zeta_q(s)=\frac{1}{1-q^{1-s}}\)

But the formula on the right hand side makes sense for greater range for values \(s\). Indeed, it is defined for almost all complex numbers \(s\) (the only points where it’s not defined is when \(q^{1-s}=1\); with complex exponentiation there are infinitely many such points, but they are quite sparse), and it is differentiable at these points, so we know that this is exactly the analytic continuation of \(\zeta_q(s)\)!

So now we simply want to study the zeros of this later function. But a reciprocal of a complex number \(1-q^{1-s}\) can never be zero! This means that the polynomial zeta function has no zeros at all. In particular, there are no zeros in the critical strip, so the polynomial Riemann hypothesis becomes vacuously true. \(\square\)

Conclusion

We see that the Riemann hypothesis for polynomials over a finite field is quite a trivial statement – we have spent more time trying to define it than to prove it, and the proof pretty much involved just working through the definitions. One might also question a number of choices we have made when defining this function. However, it can be shown that \(\zeta_q(s)\) is intimately connected to the multiplicative structure of \(\mathbb F_q[x]\), in particular – to the distribution of irreducible polynomials. To give a sense of it, define the von Mangoldt function \(\Lambda(n)\), defined for natural number \(n\), to be \(\ln p\) if \(n\) is a power of a prime \(p\) and \(\Lambda(n)=0\) otherwise. The idea behind this function is that it is the “weighted” indicator function of primes, taking into account the “average density” of primes of given order of size (the reasons we also allow prime powers are technical). Defining \(\displaystyle\psi(x)=\sum_{n\leq x}\Lambda(x)\) we then have \(\psi(x)\approx x\). The standard Riemann hypothesis, in particular lack of zeros with real part above \(\frac{1}{2}\), gives us an upper bound on the difference between the two terms similar to the bound we’ve stated in the equivalent version of Riemann hypothesis. We can try to do the same in \(\mathbb F_q[x]\). We define \(\Lambda(f)=\deg g\) for \(f\) a power of irreducible polynomial \(g\) (recall that we defined \(q^{\deg g}\) to be the “size” of \(g\), then \(\deg g\) is logarithm of the size). Since \(\zeta_q(s)\) has no zeroes at all, we expect it to give us a very good bound on the approximation of \(\displaystyle\psi_q(d)=\sum_{f\text{ monic},\deg f\leq q}\Lambda_q(f)\) and the number of monic polynomials of degree at most \(d\). Indeed, these two numbers are equal, and this fact can be deduced from \(\zeta_q(s)\) lacking zeros (though it can be also proven directly).

I will also mention the generalizations of Riemann hypothesis – there are two main extensions of the conjecture, called respectively generalized and extended Riemann hypothesis, which both state that certain functions analoguous to Riemann zeta, namely Dirichlet L-functions and Dedekind zeta functions (both discussed in another blog post of mine) have their nontrivial zeros on critical line. These conjectures also have analogues for polynomials over finite fields which are known to be true, although they are quite a bit more difficult to establish.

There are many more similarities between the world of polynomials over finite fields and (at least under standard conjectures) the world of integers. Probably the most significant difference between the two is how much easier it is to work with the former one, which hopefully this post illustrates.