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Let $M$ be a finite-dimensional simple $\mathbb Q$ algebra and $\Lambda$ an order in $M$. Its zeta-function is defined as $$ Z(s)=\sum_{I}|\Lambda/I|^{-s}, $$ where the sum runs over all left ideals and $s$ is a complex number. In a series of 14 papers in the eighties, Bushnell and Reiner claimed that the sum converges for $Re(s)>1$ and that the zeta function extends meromorphically with a pole at $s=1$. Using completely different methods, I found that in the case of dimension 4 it converges for $Re(s)>2$ only and has a pole at $s=2$. The convergence part, by the way, is easily contradicted in the case $M=M_2({\mathbb Q})$ and $\Lambda= M_2({\mathbb Z})$, the respective algebras of $2\times 2$ matrices, you simply need to restrict to the principal ideals, where already you get a sum which diverges at $s=2$.

Now, am I overlooking a renormalisation here, or have they got it wrong in all these papers? (In the second case, they probably made a mistake in the first paper and copied it along.)

In the second case, is their entire approach damaged or is it a small problem which is easily repaired?

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    $\begingroup$ Could you give some more specific references for problematic claims? It is not great to make this sort of claim without being precise enough. For what it's worth, I am looking at what seems to be their first paper on the topic (Math.Z. 173 (1980), 135-161), and for the zeta functions there (defined for arbitrary modules) the convergence is claimed when the real part is bigger than the dimension of the module (and not 1), see Prop.1 in that paper. $\endgroup$ Jul 7, 2022 at 13:53
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    $\begingroup$ @Vladimir Dotsenko: That's funny, you seem to have picked the only paper, where they claim the correct statement. Look for instance at SOLOMONS CONJECTURES AND THE LOCAL FUNCTIONAL EQUATION FOR ZETA FUNCTIONS OF ORDERS in the Bulletin or any other later paper. In total I counted 8 papers, where they give the false statement. But great, that was, what I was looking for. A single reference in which they state the correct version. Thank you! $\endgroup$
    – user473423
    Jul 7, 2022 at 15:38
  • $\begingroup$ To get Prop. 1 of their 1980 Math Z paper, Bushnell and Reiner apply the comparison test for convergence of Dirichlet series. Note that any left-module over a $\mathbb{Z}$-order $\Lambda$ is naturally a left-module over $\mathbb{Z}$. So let $L$ be any left $\Lambda$-lattice. Then the abscissa of convergence of $\zeta({}_{\Lambda} L;s)$ is no more than that of $\zeta({}_{\mathbb{Z}} L;s)$. They needed to start with some domain of convergence so that they could use an analytic method to prove Solomon's first conjecture, thus refining the domain of convergence. $\endgroup$ Oct 14, 2023 at 3:16

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It seems like you are overlooking a renormalisation. To illustrate this point, we ought to consider two cases. I'm sorry for deviating from your notation a bit. We want $A$ to be an algebra containing our order $\Lambda$, and we want $M$ to denote some other module.

First we consider the finitely generated free abelian group $M=\mathbb{Z}^{\oplus n}$ of rank $n$. Then, as a left $\mathbb{Z}$-module, $M$ has Solomon zeta function $$\zeta({}_{\mathbb{Z}}M;s)=\prod_{j=0}^{n-1}\zeta(s-j).$$

Now we consider a maximal $\mathbb{Z}$-order $\Lambda=M_n(\mathbb{Z})$ in the finite-dimensional simple $\mathbb{Q}$-algebra $A=M_n(\mathbb{Q})$. As a left $\Lambda$-module, $\Lambda$ has Solomon zeta function $$\zeta({}_\Lambda \Lambda;s)=\prod_{j=0}^{n-1}\zeta(ns-j).$$

In fact, we may see the formula $$\zeta({}_\Lambda \Lambda;s)=\zeta({}_{\mathbb{Z}}M;ns)$$ directly by Morita theory.

To be as explicit as possible, we identify $M$ with row vectors. Then we have a lattice isomorphism $$\{\text{finite index submodules of ${}_{\mathbb{Z}}M$}\}\to\{\text{finite index submodules of ${}_{\Lambda}\Lambda$}\}$$ that takes a finite index submodule $X$ of ${}_{\mathbb{Z}}M$ to the submodule $Y$ of ${}_\Lambda \Lambda$ whose rows lie in $X$. Under this correspondence, it's straightforward to check that $$|\Lambda/Y|=|M/X|^n.$$

We'll end with a small remark. You spoke about restricting to the principal left ideals of $\Lambda$. In fact, every left ideal of $\Lambda$ is principal. This follows from the fact that $\mathbb{Z}$ is a principal left ideal domain. See, for example, Theorem 17.24 of Lam's Lectures on Modules and Rings.

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