Vignéras, in her *Arithmetics of quaternion algebras*, begin chapter II.4 recalling that we know the number of integer ideals of fixed norm of a quaternion algebra $H$ over a local field $K$, ramified or not (lemma 4.1). What is of interest here is that there are $1+q+\cdots +q^n$ such ideal of fixed norm $q^n$ if $H \cong M(2,K)$. That make possible an expression of $\zeta\_H$ in terms of $\zeta\_K$.


Then, it seems that Vignéras assume $n$ is always even, writing :


$$\zeta\_H(s) = \sum\_n \sum\_{d \leqslant n} q^{d-2ns} = \sum\_{d,d'} q^{d-2(d+d')s} = \zeta\_K(2s)\zeta\_K(2s-1)$$


And here I am totally lost : why do not write 


$$\zeta\_H(s) = \sum\_n \sum\_{d \leqslant n} q^{d-ns} = \sum\_{d,d'} q^{d-(d+d')s} = \zeta\_K(s)\zeta\_K(s-1) \quad ?$$


Hence, is $n$ even ? Even if it is, what about the range of $d$, going to $n$ and not to $2n$ ?


Thanks for your help !



Factorisation of local quaternionic zeta functions
By "fixed normal" you mean "fixed norm", right? The issue is just that the norm of a (right) ideal is the square of the reduced norm $N(I)=[\mathcal{O}:I]=\mathrm{nrd}(I)^2$. 


