# History of the analytic class number formula

The (general) analytic class number formula gives a value for the residue of the Dedekind zeta function of a number field at the point $s=1$ (or, as I prefer, the leading Taylor coefficient at $s=0$). To whom should this formula be attributed?

My usual go-to place for such history questions is Narkiewicz's book Elementary and analytic theory of algebraic numbers, but it only discusses the abelian case. It lists references as late as 1929 for the result in the abelian case. Does one have to wait for Artin or Hasse?

Do you insist that the formula be interpreted as the value of a residue, hence requiring that it be known that the zeta-function of every number field is meromorphic around $s = 1$? It goes back to Dedekind that for every $K$ the limit $\lim_{s \rightarrow 1^+} (s-1)\zeta_K(s)$ exists and is given by the standard formula, but this couldn't be interpreted by him as a residue calculation since at that time it wasn't known that $\zeta_K(s)$ could be continued beyond ${\rm Re}(s) > 1$. However, if you're willing to accept the computation of that limit as a poor man's residue, then the formula goes back to Dedekind.

In 1903 Landau proved for every $K$ that $\zeta_K(s)$ can be analytically continued to ${\rm Re}(s) > 1 - 1/[K:\mathbf Q]$ (see the section on the zeta-function of a number field in Lang's Algebraic Number Theory). This was part of his proof of the prime ideal theorem, and it was the first proof for general $K$ that $\zeta_K(s)$ is meromorphic around $s = 1$. By this work Dedekind's calculation could be interpreted as a residue calculation. Therefore if you require the formula arise as a residue then in principle I'd say the result is due to Landau, although I haven't looked at his 1903 paper to see if he is explicit about that.

• Awesome. Thanks, Keith! I'm happy to call this a theorem of Dedekind then (with a mention of Landau). Did he introduce the regulator then? Seems likely. – Rob Harron Sep 10 '14 at 23:51
• See Dirichlet-Dedekind's Zahlentheorie, Section 184 (in the 11th Supplement, written by Dedekind). Letting $T(x)$ be the number of integral ideals with norm less than $x$, Dedekind computed the ratio $T(x)/x$ as $x \rightarrow \infty$. Writing $\zeta_K(s) = \sum_{n \geq 1} a_n/n^s$, we have $T(x) = \sum_{n \leq x} a_n$. From our knowledge of the pole at $s = 1$, standard methods imply $T(x) \sim \rho{x}$, where $\rho$ is the residue at $s = 1$ given by the famous formula. Dedekind proved $T(x)/x \rightarrow \rho$ by other methods and deduced $\lim_{s \rightarrow 1+} (s-1)\zeta_K(s) = \rho$. – KConrad Sep 11 '14 at 1:06
• Dedekind wrote $\zeta_K(s)$ as $\Omega(s)$ (see p. 610 -- I am looking at the Chelsea edition of Dirichlet-Dedekind). Dedekind does indeed define the regulator, on p. 597, and he uses the word Regulator. A scaled version of the regulator, denoted as $E$ (see p. 602), appears in his class number formula on p. 610 for what I am writing as $\rho$. – KConrad Sep 11 '14 at 1:11

The general formula was proved by Hecke (1917) in this paper. More precisely, the formula is an immediate consequence of equations (15a), (17a), (23a) in the paper.

Added. It seems that the residue calculation is quite a bit older (see KConrad's response), hence Hecke's new contribution was the full analytic continuation and functional equation of the Dedekind $L$-function (and the more general Hecke $L$-functions that were introduced by him).

KConrad's answer is correct, and the analytic class number formula is due to Dedekind. Yet the whole story is a little bit more complex and it is fair to say that Dedekind's analytic class number formula is essentially due to Dirichlet and Kummer, in the following sense. Dirichlet (who had already proved the class number formula for quadratic forms in his proof on the theorem of primes in arithmetic progression) determined the number of equivalence classes of forms that split into linear factors over ${\mathbb Q}(\zeta_p)$ (a key idea of which he had during a mass in the Sistine chapel), but did not publish his results as he knew that Kummer was working out a similar formula using his new language of ideal numbers. The only reason why Kummer did not (and could not) prove the general class number formula was that his theory of ideal numbers did not seem to work in general number rings. What Kummer was missing was the definition of an algebraic integer, and at this point Dedekind enters the story. He worked out a theory of ideals in algebraic rings of integers; transferring Kummer's analytic class number formula to the general case did not require any really new idea. In no way should these comments belittle Dedekind's contributions to algebraic number theory.