DetAS-X: Ideal-Counting Zeta Function Analysis

• The paper establishes that the ideal-counting zeta function arises from reducing a matrix’s minimal polynomial to a cubic form, enabling explicit Euler factorization.
• It applies a Solomon-type local-global decomposition and p-adic analysis to partition the counting problem into clearly defined valuation regions.
• The work bridges algebraic graph theory with arithmetic by linking spectral properties of matrices to the enumeration of ideals in associated Z-orders.

Searching arXiv for the specified paper to ground the article in the source record. The arXiv record (Hirasaka et al., 2016) studies the ideal-counting zeta function of the ring generated by an integral square matrix $B$ when $B$ has exactly three integral eigenvalues. The central reduction identifies the subring $(B)=\{f(B)\mid f(x)\in \mathbb{Z}[x]\}$ with a quotient of $\mathbb{Z}[x]$ by a cubic polynomial, and then normalizes the problem to the ring

$\mathbb{Z}[x]/x(x-\alpha)(x-\beta)\mathbb{Z}[x],$

where $0,\alpha,\beta\in\mathbb{Z}$ are distinct. The main objective is to compute the Dirichlet series

$\zeta_{(B)}(s)=\sum_{n\ge 1} a_n n^{-s},$

with $a_n$ equal to the number of ideals of index $n$ in $(B)$, and to make the corresponding Euler factors explicit (Hirasaka et al., 2016).

1. Algebraic formulation and normalization

For an integral square matrix $B$0, the subring $B$1 generated by $B$2 is a free $B$3-module of finite rank, so only finitely many ideals can occur at each finite index. This makes the ideal-counting Dirichlet series $B$4 well-defined (Hirasaka et al., 2016).

The key algebraic input is that when the minimal polynomial of $B$5 over $B$6 is

$B$7

with $B$8 distinct integers, one has

$B$9

A further normalization uses the isomorphism

$(B)=\{f(B)\mid f(x)\in \mathbb{Z}[x]\}$0

for any $(B)=\{f(B)\mid f(x)\in \mathbb{Z}[x]\}$1, reducing the analysis to the above form (Hirasaka et al., 2016).

This reduction is structurally important because it replaces a matrix-theoretic counting problem by a ring-theoretic one. A plausible implication is that the spectral constraint of “exactly three integral eigenvalues” is valuable not only for graph-theoretic examples but also because it places the ambient ring in a tractable cubic quotient framework.

2. Global zeta factorization and the role of bad primes

A Solomon-type local-global decomposition is used. If $(B)=\{f(B)\mid f(x)\in \mathbb{Z}[x]\}$2 is a semisimple $(B)=\{f(B)\mid f(x)\in \mathbb{Z}[x]\}$3-algebra, $(B)=\{f(B)\mid f(x)\in \mathbb{Z}[x]\}$4 a $(B)=\{f(B)\mid f(x)\in \mathbb{Z}[x]\}$5-order in $(B)=\{f(B)\mid f(x)\in \mathbb{Z}[x]\}$6, and $(B)=\{f(B)\mid f(x)\in \mathbb{Z}[x]\}$7 a $(B)=\{f(B)\mid f(x)\in \mathbb{Z}[x]\}$8-lattice, then there are local rational functions $(B)=\{f(B)\mid f(x)\in \mathbb{Z}[x]\}$9 such that

$\mathbb{Z}[x]$0

where $\mathbb{Z}[x]$1 is the finite set of primes at which $\mathbb{Z}[x]$2 is not maximal, and $\mathbb{Z}[x]$3 is the ambient zeta factor coming from the semisimple algebra decomposition (Hirasaka et al., 2016).

In the cubic quotient under consideration,

$\mathbb{Z}[x]$4

the bad primes are exactly the primes dividing

$\mathbb{Z}[x]$5

Moreover,

$\mathbb{Z}[x]$6

Hence

$\mathbb{Z}[x]$7

(Hirasaka et al., 2016).

This factorization isolates all nontrivial arithmetic in finitely many Euler factors. For primes $\mathbb{Z}[x]$8, the local factor is simply $\mathbb{Z}[x]$9, so away from the bad set the ideal-counting behavior is the same as that of $\mathbb{Z}[x]/x(x-\alpha)(x-\beta)\mathbb{Z}[x],$0. This suggests that the singular local structure of the cubic quotient is concentrated exactly where the three roots $\mathbb{Z}[x]/x(x-\alpha)(x-\beta)\mathbb{Z}[x],$1 fail to remain pairwise distinct modulo $\mathbb{Z}[x]/x(x-\alpha)(x-\beta)\mathbb{Z}[x],$2.

3. Local model over $\mathbb{Z}[x]/x(x-\alpha)(x-\beta)\mathbb{Z}[x],$3

The local counting problem is formulated over

$\mathbb{Z}[x]/x(x-\alpha)(x-\beta)\mathbb{Z}[x],$4

The authors fix the $\mathbb{Z}[x]/x(x-\alpha)(x-\beta)\mathbb{Z}[x],$5-basis

$\mathbb{Z}[x]/x(x-\alpha)(x-\beta)\mathbb{Z}[x],$6

of $\mathbb{Z}[x]/x(x-\alpha)(x-\beta)\mathbb{Z}[x],$7, and parametrize every full ideal $\mathbb{Z}[x]/x(x-\alpha)(x-\beta)\mathbb{Z}[x],$8 as a free $\mathbb{Z}[x]/x(x-\alpha)(x-\beta)\mathbb{Z}[x],$9-module of rank $0,\alpha,\beta\in\mathbb{Z}$0 with basis

$0,\alpha,\beta\in\mathbb{Z}$1

for some $0,\alpha,\beta\in\mathbb{Z}$2 and $0,\alpha,\beta\in\mathbb{Z}$3 (Hirasaka et al., 2016).

The triple $0,\alpha,\beta\in\mathbb{Z}$4 is uniquely determined by the ideal and serves as its type. The index of such an ideal is

$0,\alpha,\beta\in\mathbb{Z}$5

The ideal condition $0,\alpha,\beta\in\mathbb{Z}$6 becomes the congruence system

$0,\alpha,\beta\in\mathbb{Z}$7

$0,\alpha,\beta\in\mathbb{Z}$8

$0,\alpha,\beta\in\mathbb{Z}$9

The counting problem is then converted into a sum over admissible valuation data and residue parameters (Hirasaka et al., 2016).

The local behavior depends on the $\zeta_{(B)}(s)=\sum_{n\ge 1} a_n n^{-s},$0-adic valuations $\zeta_{(B)}(s)=\sum_{n\ge 1} a_n n^{-s},$1, $\zeta_{(B)}(s)=\sum_{n\ge 1} a_n n^{-s},$2, and $\zeta_{(B)}(s)=\sum_{n\ge 1} a_n n^{-s},$3, where for $\zeta_{(B)}(s)=\sum_{n\ge 1} a_n n^{-s},$4, $\zeta_{(B)}(s)=\sum_{n\ge 1} a_n n^{-s},$5 denotes the nonnegative integer such that $\zeta_{(B)}(s)=\sum_{n\ge 1} a_n n^{-s},$6, and $\zeta_{(B)}(s)=\sum_{n\ge 1} a_n n^{-s},$7. By symmetry, one may assume

$\zeta_{(B)}(s)=\sum_{n\ge 1} a_n n^{-s},$8

in the local analysis (Hirasaka et al., 2016).

4. Parametrization of admissible coefficients

Two auxiliary sets organize the local counting. First,

$\zeta_{(B)}(s)=\sum_{n\ge 1} a_n n^{-s},$9

denotes the set of $a_n$0 satisfying (5). Second,

$a_n$1

denotes the set of pairs $a_n$2 satisfying (6) and (7). The number of ideals of type $a_n$3 is therefore

$a_n$4

(Hirasaka et al., 2016).

A decisive lemma gives an explicit description of $a_n$5. Writing $a_n$6,

$a_n$7

This is the first major case split in the argument (Hirasaka et al., 2016).

For a fixed $a_n$8, the allowable values of $a_n$9 are described by

$n$0

and the number of solutions in $n$1 is

$n$2

Once $n$3 is fixed, the count of admissible $n$4 is therefore a pure power of $n$5 (Hirasaka et al., 2016).

Further lemmas refine this by splitting $n$6 according to the valuation of $n$7, computing $n$8 in the relevant regimes, and decomposing into sets $n$9 where $(B)$0 with exact cardinalities. This is what permits explicit Euler factors rather than only an algorithmic procedure (Hirasaka et al., 2016).

5. Regionwise summation and local generating functions

The local ideal-counting function is defined by

$(B)$1

With the notation

$(B)$2

this becomes a rational-function computation in $(B)$3 (Hirasaka et al., 2016).

Theorem 3.7 partitions the $(B)$4-plane into six regions

$(B)$5

according to the relative sizes of $(B)$6, and $(B)$7 has a uniform closed form on each region (Hirasaka et al., 2016).

Among the explicit cases proved are the following:

• If $(B)$8, then

$(B)$9

• If $B$00 and $B$01, then

$B$02

The delicate finite region $B$03 requires the finer decomposition from Lemma 2.11 (Hirasaka et al., 2016).

This regionwise organization is methodologically significant because it converts a valuation-sensitive ideal classification into a finite family of geometric-series evaluations. A plausible implication is that the principal technical obstacle is not the final summation but the precise stratification of $B$04-adic parameter space.

6. Explicit formulas and graph-theoretic instances

The global zeta function always takes the form

$B$05

with $B$06 determined explicitly by the valuations $B$07, $B$08, and $B$09 (Hirasaka et al., 2016).

Several valuation patterns are treated in closed form.

Valuation pattern	Explicit formula
$B$10	$B$11
$B$12	$B$13
$B$14	$B$15

For the pattern $B$16, the paper states the formula in Proposition 4.1 in equivalent simplified forms. The listed expression follows the closed form given there (Hirasaka et al., 2016).

A prominent example is the Petersen graph. Its adjacency matrix has eigenvalues $B$17, and after shifting the corresponding ring is

$B$18

Here the relevant valuations satisfy

$B$19

so Proposition 4.1 gives

$B$20

(Hirasaka et al., 2016).

The paper also states that, for the adjacency matrix of the cycle of length $B$21, a formula is obtained from Proposition 4.3 and matches a computer-assisted earlier computation. It further remarks that the same framework applies to many graph adjacency matrices with exactly three integral eigenvalues, including

$B$22

provided the local valuation pattern matches one of the treated cases (Hirasaka et al., 2016).

7. Multiplicativity and mathematical significance

Because the zeta function is an Euler product, the coefficients $B$23 are multiplicative. Equivalently,

$B$24

with each $B$25 a rational function in $B$26, so the coefficients are determined by the counts at prime powers together with multiplicativity across coprime $B$27 (Hirasaka et al., 2016).

For primes $B$28, the local factor is

$B$29

and the nontrivial arithmetic is localized at finitely many primes. This sharply separates generic behavior from singular local corrections (Hirasaka et al., 2016).

The overall derivation proceeds by algebraic reduction to the cubic quotient, localization at $B$30, parametrization of full ideals by a triangular $B$31-basis, translation of closure under multiplication by $B$32 into congruences (5)–(7), valuation-theoretic analysis of the solution sets, partition of the $B$33-plane into six regions, summation of the resulting geometric series, and assembly of the Euler product (Hirasaka et al., 2016). In methodological terms, the work connects spectral data of integral matrices—particularly adjacency matrices with three integral eigenvalues—to explicit ideal zeta functions of associated $B$34-orders. This suggests a concrete bridge between algebraic graph theory, the arithmetic of orders, and local counting methods over $B$35.