One of the recurring themes of modern number theory seems to be a nagging feeling that almost all the things we know (and many more things we don’t know) can be expressed as elementary properties of {\zeta} and {L}-functions.

For example, the zeta function of a number field {K} is defined by the product

\displaystyle \zeta_K(s) = \prod_{\mathfrak{p} \text{ prime ideal of } \mathcal{O}_k} (1 - (N\mathfrak{p})^{-s})^{-1}.
If we expand this as a Dirichlet series

\displaystyle \zeta_K(s) = \sum_n a_n n^{-s}
the coefficients {a_n} give detailed information about the factorisation of ideals in {K}. For example, if {a_p=1} then {p} is totally ramified, whereas if it is equal to the degree of the field, {p} is totally split.

It is also well-known that the Riemann zeta function (the case {K={\mathbb Q}} above) controls the distribution of the prime numbers in {{\mathbb Z}}. By modifying this {\zeta} function we can obtain the Dirichlet L-functions which give us even more information. To define these, we start with a homomorphism {\chi: {\mathbb Z} \rightarrow {\mathbb Z}/q{\mathbb Z} \rightarrow {\mathbb C}^*} called a Dirichlet character, and `twist’ the {\zeta}-function using it:

\displaystyle L(\chi, s) = \sum_n \chi(n) n^{-s} = \prod_p (1 - \chi(p)p^{-s})^{-1}.
For example, the trivial character {\chi_1 = 1} recovers the original {\zeta}-function. Dirichlet famously used this, together with observations about the (lack of) zeroes of these (mostly) convergent power series at {Re(s)>1} and {s=1}, to deduce that any arithmetic progression which might plausibly contain infinitely many primes does. Note that if {q} is the least quotient {\chi} factors through, it is called the conductor of {\chi}.

It seems to be true that certainly for {E/{\mathbb Q}} an abelian extension, the above two objects are related, and in fact {\zeta_E} can be written as a product of Dirichlet L-functions. I don’t know enough class field theory to go into the general cases, but want to look at a specific case that seems quite interesting. Apparently the following result is true. Let {q} be a fixed prime, and {\chi} any quadratic (taking values {1} and {-1}) character with conductor {q}. Now consider the quadratic field {K = {\mathbb Q}(\sqrt{\chi(-1) q})}. We have the following identity:

\displaystyle \zeta_K(s) = \zeta(s) L(\chi, s).

Looks innocuous enough…. surely even if this is true, it can’t say much interesting.

Well, let’s unravel what it says:

\displaystyle \prod_{p \text{ ramified}} (1-p^{-s})^{-1} \prod_{p \text{ split}} (1-p^{-s})^{-2} \prod_{p \text{ inert}} (1-p^{-2s})^{-1} = \zeta(s) L(\chi, s).
So cancelling the familiar factors from {\zeta}, it says that

\displaystyle L(\chi, s) = \prod_{p \text{ split}} (1-p^{-s})^{-1} \prod_{p \text{ inert}} (1+p^{-s})^{-1}.
So comparing product expansions, it is equivalent to

\displaystyle \chi(p) = \begin{cases} 1 \text{ iff }p\text{ is split,}\\ 0 \text{ iff }p\text{ is ramified,}\\ -1 \text{ iff }p\text{ is inert.} \end{cases}.

Interesting. This splitting into cases reminds us of a more classical area of algebraic number theory. Write {K={\mathbb Q}(\sqrt{d})}, and assume {d \equiv 1 \mod 4}. Then the minimal polynomial is {X^2 + X + (d-1)/4}, which factorises modulo an odd prime {p} iff it does after we’ve multiplied by 4 and completed the square to get {(2X+1)^2 + d}, whose factorisation modulo {p}, together with the Kummer-Dedekind theorem for the splitting of primes in number fields whose ring of integers is primitively generated, gives a similar criterion in terms of the Legendre symbol:

\displaystyle \left(\frac{d}{p}\right) = \begin{cases} 1 \text{ iff }p\text{ is split,}\\ 0 \text{ iff }p\text{ is ramified,}\\ -1 \text{ iff }p\text{ is inert.} \end{cases}.
Now, take {\chi(n) = \left(\frac{n}{q}\right)} and, noting that {\chi(-1) = (-1)^{(q-1)/2}}, we see straight away, comparing the above two statements, that:

\displaystyle \left(\frac{p}{q}\right) = \left(\frac{d}{p}\right) = \left(\frac{(-1)^{(q-1)/2}}{p}\right)\left(\frac{q}{p}\right) = (-1)^{(p-1)(q-1)/4}\left(\frac{q}{p}\right).

So our identity in terms of {\zeta} and L-functions was actually a (fairly heavily) disguised statement of Gauss’ famous law of quadratic reciprocity! Of course, I haven’t given you a proof: I’m not personally sure exactly how to prove the identity about the {\zeta} and L-functions. Maybe I’ll let you know when I figure it out, but in the meantime there are plenty of proofs of quadratic reciprocity out there: http://www.rzuser.uni-heidelberg.de/~hb3/rchrono.html. I should also refer the interested reader to Kolowski’s opening chapter of `an introduction to the Langlands program’ which was certainly where I read about this.