Define numbers $H_k$ for integers $k\geq 4$ by $\sum_{x \in \mathbf{Z}[i]}x^{-k}=\frac{H_k}{k!} \omega^k$, where $\omega=\frac{\Gamma(1/4)^2}{\sqrt{2\pi}}$. These are nonzero when $4|k$, and Hurwitz proved that they are rational numbers. Numerical experiments quickly reveal some remarkable properties of these numbers. Here is a table of the $H_k$'s for $4\leq k \leq 80$; I have factored the numerators and bolded all of their factors which are $\equiv 1 \; \mathrm{mod}\; 4$:

alt text http://www2.bc.edu/~hansendd/hurwitz.png

First of all, it seems the denominator of $H_k$ is given as the product of all primes $p$ such that $(p-1) \mid k$ and either $p=2$ or $p\equiv 1 \; \mathrm{mod} \; 4$. This is in obvious analogy to the von Staudt-Clausen theorem. More surprisingly, it seems the numerator is divisible by *every* prime $p$ with $p < k-4$ and $p \equiv 3 \; \mathrm{mod} \; 4$, with fairly regular exponents. This stands in marked contrast to Bernoulli numbers, whose numerators display no patters nearly so obvious. Are either of these observations proven somewhere? Do the inert primes dividing the numerator have arithmetic significance? (The sporadic split primes dividing the numerators are known to have arithmetic significance; see the paper "Kummer's criterion for Hurwitz numbers" by Coates and Wiles.)

Edit: The denominator given above for $H_{80}$ should by $6970$. In case you want to play along, I've been calculating these in Mathematica using the code:

S[k_]:=2*(2*Pi)^k*(-BernoulliB[k]/(2*k)+Sum[DivisorSigma[k-1,n]*Exp[-2*Pi*n],{n,1,800}])/(k-1)!

Om:=Gamma[1/4]^2/Sqrt[2*Pi]

Hur[j_]:=RootApproximant[N[S[j]*j!/Om^j,240],1]

(Adjust the 240 and 800 accordingly; it gives wrong answers with e.g. 800 replaced by infinity!)

5more comments