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?



History of the analytic class number formula
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. 


