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arxiv: 2604.03108 · v1 · submitted 2026-04-03 · 🧮 math.RT

On Zeta functions and μ-series of string algebras

Rohun Easwar , Amit Kuber , Mihir Mittal This is my paper

Pith reviewed 2026-05-13 18:49 UTC · model grok-4.3

classification 🧮 math.RT MSC 16G2016G60
keywords string algebrasμ-serieszeta functionsdomestic algebrasexponential growthtame representation typebandsmodule growth
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The pith

A string algebra is domestic if and only if its μ-series is rational.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper defines the μ-series of a tame algebra as the generating function that counts the smallest number of one-parameter families of modules of each total dimension. For string algebras it introduces a zeta function built as an Euler product over the bands, then proves an analogue of the prime-number theorem that relates the growth of bands to the growth of modules. This relation is used to show that the number of indecomposable modules grows exponentially precisely when the algebra is non-domestic. The same relation yields the main equivalence: the μ-series is a rational function exactly when the algebra is domestic.

Core claim

For a string algebra Λ the zeta function is the infinite product ∏_b (1 − t^{|b|})^{-1} taken over bands b up to cyclic permutation. An analogue of the prime-number theorem for this zeta function implies that the module growth is exponential unless only finitely many bands exist. Consequently the minimal number of one-parameter families μ_Λ(m) satisfies that the generating series ∑ μ_Λ(m) t^m is a rational function if and only if Λ is domestic.

What carries the argument

The zeta function ζ_Λ(t) = ∏_{b ∈ Ba(Λ)} (1 − t^{|b|})^{-1} whose poles and growth encode the bands and thereby control the rationality of the μ-series.

If this is right

  • Non-domestic string algebras have exponential module growth.
  • The μ-series of a string algebra is rational precisely when the algebra is domestic.
  • The growth of bands obeys an analogue of the prime-number theorem.
  • Domesticity can be read off from the rationality of a single generating function.
  • The zeta product over bands completely determines the asymptotic module count.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Rationality of the μ-series supplies an algorithmic criterion that decides domesticity for any given string algebra once its bands are listed.
  • The same zeta-function construction may distinguish domestic from non-domestic behaviour in other classes of tame algebras that admit a band description.
  • The arithmetic analogy with the classical prime-number theorem raises the possibility that further number-theoretic invariants of the zeta function, such as special values, also carry representation-theoretic meaning.
  • If the rationality test extends beyond string algebras it would give a uniform way to classify tame algebras by the algebraic degree of their module-counting series.

Load-bearing premise

The bands up to cyclic permutation and the minimal counts μ_Λ(m) of one-parameter families are assumed to be the exact data that determine module growth for string algebras.

What would settle it

Exhibit a concrete non-domestic string algebra for which the power series ∑ μ_Λ(m) t^m is nevertheless a rational function, or exhibit a domestic string algebra whose series is irrational.

Figures

Figures reproduced from arXiv: 2604.03108 by Amit Kuber, Mihir Mittal, Rohun Easwar.

Figure 1
Figure 1. Figure 1: An example of a directed graph Remark 2.4. For an (n × n)-matrix A with characteristic polynomial χA, we have det(I − tA) = t nχA(t −1 ). For a matrix A, let Spec A denote its spectrum (multiset of eigenvalues with multiplicity) and R(A) denote its spectral radius (maximum absolute value of an eigenvalue of A). Definition 2.5. An (n × n) real matrix A is said to be irreducible if there does not exist a per… view at source ↗
Figure 2
Figure 2. Figure 2: GP2,3 with ρ = {a 2 , b3 , ab, ba} bA b 2 B2 Ba Ab aB bbA Abb AbA bAb BBa aBB aBa BaB [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
read the original abstract

Let $\overline\mu_\Lambda(t):=\sum\limits_{m\geq1}\mu_\Lambda(m)t^m$ be the \emph{$\mu$-series} of a finite-dimensional tame algebra $\Lambda$ over an algebraically closed field, where $\mu_\Lambda(m)$ denotes the minimal number of one-parameter families of $\Lambda$-modules with total dimension $m$. When $\Lambda$ is a string algebra with $\mathrm{Ba}(\Lambda)$ as its set of bands up to cyclic permutation, define the \emph{zeta function} $\zeta_\Lambda(t):=\prod\limits_{\mathfrak b\in\mathrm{Ba}(\Lambda)}(1-t^{|\mathfrak b|})^{-1}$, where $|\mathfrak b|$ is the length of $\mathfrak b$. We prove an analogue of the prime number theorem for string algebras and use it to conclude that non-domestic string algebras are of exponential growth. Finally, we show that a string algebra is domestic if and only if its $\mu$-series is rational.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 3 minor

Summary. The paper defines the μ-series of a tame algebra Λ as the generating function ∑ μ_Λ(m) t^m, where μ_Λ(m) is the minimal number of one-parameter families of modules of total dimension m. For string algebras it introduces the zeta function ζ_Λ(t) = ∏_{b ∈ Ba(Λ)} (1 - t^{|b|})^{-1} over bands up to cyclic permutation. It proves an analogue of the prime-number theorem for these zeta functions, deduces that non-domestic string algebras have exponential growth, and establishes that a string algebra is domestic if and only if its μ-series is rational.

Significance. If the results hold, the work supplies an analytic criterion (rationality of the μ-series) that distinguishes domestic from non-domestic string algebras and yields precise growth rates via a Tauberian extraction from the pole of the zeta function at t=1. The approach rests on standard facts about bands and one-parameter families, producing parameter-free derivations that link representation-theoretic growth directly to the analytic properties of an Euler product; this could furnish new tools for the classification of tame algebras.

minor comments (3)
  1. [§1] §1 (Introduction): the sentence defining μ_Λ(m) as the 'minimal number' should explicitly reference the standard parametrization of bands by cyclic permutation to avoid any ambiguity in the count.
  2. [Theorem on rationality] The proof of the rationality statement for infinite bands invokes the natural boundary of a lacunary series; a brief remark on why this precludes a linear recurrence would help readers unfamiliar with the relevant Tauberian or complex-analysis facts.
  3. [Notation] Notation: the overline on μ̄_Λ(t) is used in the abstract but not consistently introduced in the main text; adopt a single symbol throughout.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary and significance assessment of our work establishing an analytic criterion for domestic string algebras via rationality of the μ-series. We appreciate the recommendation for minor revision. No major comments were provided in the report, so we have no specific points to address point-by-point.

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained from standard facts

full rationale

The paper starts from the standard definition of bands up to cyclic permutation and the count μ_Λ(m) as the number of bands whose length divides m. It then defines the zeta function explicitly as the Euler product ∏ (1 - t^|b|)^(-1) and the μ-series as the corresponding sum ∑ t^|b| / (1 - t^|b|). The rationality statement for domestic algebras is immediate from the finite product being rational. The non-domestic case uses the lacunary series having the unit circle as natural boundary (a standard analytic fact) to conclude non-rationality. The prime-number-theorem analogue follows by Tauberian extraction from the simple pole at t=1, directly implying exponential growth. No equation reduces to a fitted parameter renamed as prediction, no self-citation is load-bearing for the central claims, and no ansatz is smuggled in; all steps are explicit constructions or applications of external analytic/number-theoretic tools.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 2 invented entities

Relies on standard domain assumptions from representation theory; introduces new generating functions without additional free parameters or invented entities beyond the definitions themselves.

axioms (2)
  • domain assumption Finite-dimensional tame algebras over algebraically closed fields have well-defined μ_Λ(m) as the minimal number of one-parameter families of modules of dimension m
    Foundation for the μ-series definition and growth statements.
  • domain assumption Bands of a string algebra are considered up to cyclic permutation
    Directly used to index the product in the zeta function definition.
invented entities (2)
  • zeta function ζ_Λ(t) no independent evidence
    purpose: Product over bands to encode lengths for analytic growth analysis
    Newly defined in the paper for string algebras.
  • μ-series no independent evidence
    purpose: Generating function summing minimal one-parameter family counts
    Defined in the paper to capture module complexity.

pith-pipeline@v0.9.0 · 5470 in / 1403 out tokens · 66849 ms · 2026-05-13T18:49:46.906739+00:00 · methodology

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Reference graph

Works this paper leans on

4 extracted references · 4 canonical work pages

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