Bernoulli Numbers, Fermat’s Last Theorem, and the Riemann Zeta Function

The Bernoulli numbers are the Taylor series coefficients of the function

$\displaystyle \frac{x}{e^{x}-1}$ .

The $n$ -th Bernoulli number $B_{n}$ is zero for odd $n$ , except for $n=1$ , where it is equal to $-1/2$ . For the first few even numbers, we have

$\displaystyle B_0=1,\; B_{2}=\frac{1}{6}, \; B_{4}=-\frac{1}{30}, \; B_6=\frac{1}{42}, \; B_{8}=-\frac{1}{30}, \; B_{10}=\frac{5}{66}$ .

Bernoulli numbers have many interesting properties, and many mathematicians have studied them for a very long time. They are named after Jacob Bernoulli, but were also studied by Seki Takakazu in Japan at around the same time (end of 17th/beginning of 18th century). In this post I want to focus more on the work of Ernst Edouard Kummer, more than a century after Bernoulli and Takakazu.

We’re going to come back to Bernoulli numbers later, but for now let’s talk about something completely different – Fermat’s Last Theorem, which Kummer was working on. In the time of Kummer, a proposal to study Fermat’s Last Theorem by factoring both sides of the famous equation into linear terms. Just as $x^2+y^2$ factors into

$\displaystyle x^2+y^2=(x+iy)(x-iy)$ ,

we would have that $x^{p}+y^{p}$ also factors into

$\displaystyle x^{p}+y^{p}=(x+\zeta_{p}y)(x+\zeta_{p}^{2} y)...(x+\zeta_{p}^{p-1} y)$

where $\zeta_{p}$ is a $p$ -th root of unity.

However, there is a problem. In these kinds of numbers where $p$ -th roots of unity are adjoined, factorization may not be unique! Hence Kummer developed the theory of “ideals” to study this (see also The Fundamental Theorem of Arithmetic and Unique Factorization).

Unique factorization does not work with the numbers themselves, but it works with ideals (this is true for number fields, since they form what is called a “Dedekind domain”). Hence the original name of ideals was “ideal numbers”. To number fields we associate an “ideal class group“. If this group has only one element, unique factorization holds. If not, then things can get complicated. The ideal class group (together with the Galois group) is probably the most important group in number theory.

Kummer found that if $p$ is a “regular prime“, i.e. if p does not divide the number of elements of the ideal class group (also known as the class number) of the “ $p$ -th cyclotomic field” (the rational numbers with $p$ -th roots of unity adjoined), then Fermat’s Last Theorem is true for $p$ .

Let’s go back to Bernoulli numbers now – Kummer also found that a prime $p$ is regular if and only if it does not divide the numerator for the nth Bernoulli number, for all $n$ less than $p-1$ . In other words, Kummer proved Fermat’s Last Theorem for prime exponents not dividing the numerators of Bernoulli numbers! Fermat’s Last Theorem has now been proved in all cases, but the work of Kummer remains influential.

So we’ve related Bernoulli numbers to ideal class groups and the very famous Fermat’s Last Theorem. Now let us relate Bernoulli numbers to another very famous thing in math – the Riemann zeta function (see also Zeta Functions and L-Functions).

It is known that the Bernoulli numbers are related to values of the Riemann zeta function at the negative integers (so we need the analytic continuation to do this) by the following equation: $B_n=n \zeta(1-n)$ for $n$ greater than or equal to $1$ .

Now, Kummer also discovered that Bernoulli numbers satisfy certain congruences modulo powers of a prime $p$ , in particular

$\displaystyle \frac{B_{m}}{m}\equiv \frac{B_{n}}{n} \mod p$

where $m$ and $n$ are positive even integers neither of which are divisible by $(p-1)$ , and $m\equiv n \mod (p-1)$ . Here congruence for two rational numbers $\frac{a}{b}$ and $\frac{c}{d}$ means that $ad$ is congruent to $cd$ mod $p$ .

We also have a more general congruence for bigger powers of $p$ :

$\displaystyle (1-p^{m-1})\frac{B_{m}}{m}\equiv (1-p^{n-1})\frac{B_{n}}{n} \mod p^{a+1}$

where $m$ and $n$ are positive even integers neither of which are divisible by $(p-1)$ , and $m\equiv n \mod \varphi(p^{a}+1)$ , $\varphi^{a}+1$ being the number of positive integers less than $p^{a+1}$ that are also mutually prime to it.

By by our earlier discussion, this means the special values of the Riemann zeta function also satisfy congruences modulo powers of $p$ .

Congruences modulo powers of $p$ is encoded in modern language by the “ $p$ -adic numbers” (see also Valuations and Completions) introduced by Kurt Hensel near the end of the 19th century. The congruences between the special values of the Riemann zeta function is now similarly encoded in a $p$ -adic analytic function known as the Kubota-Leopoldt $p$ -adic L-function.

So again, to summarize the story so far – Bernoulli numbers are related to the ideal class group and also to the special values of the Riemann zeta function, and bridge the two subjects.

If this reminds you of the analytic class number formula, well in fact that is one of the ingredients in the proof of Kummer’s result relating regular primes and the Bernoulli numbers. Moreover, the information that they encode is related to divisibility or congruence modulo primes or their powers. This is where the $p$ -adic L-functions come in.

The Bernoulli numbers also appear in the constant term of the Fourier expansion of Eisenstein series. The Eisenstein series is an example of a modular form (see also Modular Forms), which gives us Galois representations. The Galois group, on the other hand is related to the ideal class group by class field theory (see also Some Basics of Class Field Theory). So this is one way to create the bridge between the two concepts. In fact, this was used to prove the Herbrand-Ribet theorem, a stronger version of Kummer’s result.

So we also have modular forms in the picture. In modern research all of these are deeply intertwined – ideal class groups, zeta functions, congruences, and modular forms.

References:

Bernoulli number on Wikipedia

Riemann zeta function on Wikipedia

Kummer’s congruence on Wikipedia

p-adic L-function on Wikipedia

Herbrand-Ribet theorem on Wikipedia

Bernoulli numbers, Hurwitz numbers, p-adic L-functions and
Kummer’s criterion by Alvaro Lozano-Robledo

An introduction to Eisenstein measures by E. E. Eischen

How can we construct abelian Galois extensions of basic number
fields? by Barry Mazur