Linear recurrences and rational Lambert series

Igor Rivin

arxiv: 2604.25151 · v1 · submitted 2026-04-28 · 🧮 math.NT

Linear recurrences and rational Lambert series

Igor Rivin This is my paper

Pith reviewed 2026-05-07 15:25 UTC · model grok-4.3

classification 🧮 math.NT

keywords linear recurrencesLambert seriesrational generating functionsfinitely supported sequencesdivisor sumsnumber theory

0 comments

The pith

If a sequence is eventually linearly recurrent and its Lambert series is rational, then the sequence must be finitely supported.

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

The paper proves that for any sequence γ that is eventually linearly recurrent, rationality of the associated Lambert series L_γ(z) forces γ to have only finitely many nonzero terms. This is presented equivalently as the claim that the only sequences whose ordinary generating function is rational and whose Lambert series is also rational are the finitely supported ones. The argument reduces the problem by specializing the values of γ at a finite place of a finitely generated ring containing them, after which the linear recurrence becomes periodic over the resulting finite field. Periodicity then implies that an infinite-support sequence would produce a non-rational Lambert series, yielding the desired contradiction.

Core claim

If γ is eventually linearly recurrent and L_γ(z) is rational, then γ is finitely supported. Equivalently, among sequences with rational ordinary generating function, the only ones whose Lambert series is rational are the finitely supported sequences. The proof specializes the data at a finite place of a finitely generated ring and then uses the periodicity of recurrences over finite fields.

What carries the argument

Specialization of the sequence values at a finite place of a finitely generated ring, reducing the eventual linear recurrence to a periodic sequence over a finite field.

If this is right

Sequences whose ordinary generating function is rational and whose Lambert series is rational must be finitely supported.
No infinite eventually linearly recurrent sequence can have a rational Lambert series.
Rationality of both the ordinary generating function and the Lambert series is a strong constraint that collapses to finite support under the recurrence hypothesis.

Where Pith is reading between the lines

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

The result constrains possible rationality properties for divisor-sum generating functions attached to recurrent sequences arising in arithmetic contexts.
It suggests that similar specialization arguments could be applied to other rationality questions involving Lambert-type series or their generalizations.
Explicit checks over small finite fields for periodic sequences with infinite support would confirm that their Lambert series are always non-rational.

Load-bearing premise

The sequence takes values in a finitely generated ring so that specialization at a finite place is possible and the recurrence remains periodic over the resulting finite field.

What would settle it

An eventually linearly recurrent sequence with infinitely many nonzero terms whose associated Lambert series L_γ(z) is nevertheless a rational function.

read the original abstract

For a sequence $\gamma=(\gamma_n)_{n\ge 1}$, define \[ L_\gamma(z):=\sum_{n\ge 1}\gamma_n\frac{z^n}{1-z^n} =\sum_{n\ge 1}\Bigl(\sum_{d\mid n}\gamma_d\Bigr)z^n. \] We prove a short rigidity theorem: if $\gamma$ is eventually linearly recurrent and $L_\gamma(z)$ is rational, then $\gamma$ is finitely supported. Equivalently, among sequences with rational ordinary generating function, the only ones whose Lambert series is rational are the finitely supported sequences. The proof specializes the data at a finite place of a finitely generated ring and then uses the periodicity of recurrences over finite fields.

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.

Desk Editor's Note private letter to a colleague

The paper proves a clean rigidity result that eventually linearly recurrent sequences with rational Lambert series must be finitely supported, but the theorem statement omits the finitely generated ring hypothesis that the proof actually requires.

read the letter

The main takeaway is that this short note gives a rigidity theorem: if γ is eventually linearly recurrent and its Lambert series is rational, then γ has finite support. Equivalently, among sequences with rational ordinary generating functions, only the finitely supported ones have rational Lambert series. The proof reduces to finite fields via specialization at a finite place and uses the resulting periodicity of the recurrence there to reach a contradiction with rationality unless support is finite. That direction is new and does not collapse to earlier work on ordinary generating functions alone. The argument is direct and relies on a standard technique rather than any heavy machinery, which keeps the paper short and readable. The central claim holds up once the ring hypothesis is in place, and there is no circularity or parameter fitting. A clear soft spot is that the abstract states the theorem for an arbitrary sequence without mentioning that the values and coefficients must lie in a finitely generated ring. The specialization step fails if the ring has no finite places, for instance when the γ_n generate a transcendental extension of Q. The proof sketch acknowledges the ring but the theorem statement does not, so the claimed generality is not fully supported by the argument as written. This is fixable by adding the hypothesis, but it is a real omission in the current version. The paper is aimed at number theorists who work with generating functions, recurrences, and Lambert series. A reader already familiar with specialization arguments will see the value quickly and can check the details in a few pages. It is focused enough that a serious referee should look at it to confirm the reduction works in the intended setting and to suggest the needed clarification on the ring. I would send it to peer review rather than desk reject.

Referee Report

1 major / 0 minor

Summary. The paper proves a rigidity theorem: if a sequence γ is eventually linearly recurrent and its Lambert series L_γ(z) = ∑ γ_n z^n / (1 - z^n) is rational, then γ is finitely supported. Equivalently, among sequences with rational ordinary generating functions, only finitely supported ones have rational Lambert series. The proof specializes the sequence values and recurrence coefficients at a finite place of the finitely generated ring they generate, reducing to a finite field in which the recurrence forces eventual periodicity, yielding a contradiction with rationality of L_γ(z) unless γ has finite support.

Significance. If the result holds, it supplies a clean algebraic rigidity statement connecting eventual linear recurrence with rationality of the Lambert series (which encodes the divisor-sum function). The argument draws on a standard and effective technique—specialization to finite places followed by periodicity of linear recurrences over finite fields—which is a clear strength and keeps the proof short. The result would be of interest in number theory and combinatorics on words, provided the hypotheses are stated precisely.

major comments (1)

[Abstract] Abstract (and presumably the main theorem statement): the claim is formulated for an arbitrary sequence γ that is eventually linearly recurrent, without the hypothesis that the values γ_n and the recurrence coefficients lie in a finitely generated ring. The proof sketch relies on this hypothesis to perform specialization at a finite place and reduce to a finite field where periodicity applies. Without it the argument does not cover sequences whose values generate rings without finite places (e.g., transcendental extensions of ℚ), so the stated generality exceeds what the given reasoning establishes. The hypothesis must be added to the theorem statement.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and for identifying the missing hypothesis in the theorem statement. We agree that the generality as stated exceeds the proof and will revise accordingly.

read point-by-point responses

Referee: [Abstract] Abstract (and presumably the main theorem statement): the claim is formulated for an arbitrary sequence γ that is eventually linearly recurrent, without the hypothesis that the values γ_n and the recurrence coefficients lie in a finitely generated ring. The proof sketch relies on this hypothesis to perform specialization at a finite place and reduce to a finite field where periodicity applies. Without it the argument does not cover sequences whose values generate rings without finite places (e.g., transcendental extensions of ℚ), so the stated generality exceeds what the given reasoning establishes. The hypothesis must be added to the theorem statement.

Authors: We agree with this assessment. The proof relies on specializing at a finite place of the finitely generated ring generated by the sequence values and recurrence coefficients, which is not available in general (e.g., for transcendental extensions). We will revise the abstract and the main theorem to explicitly include the hypothesis that γ_n and the recurrence coefficients lie in a finitely generated ring over ℤ (or ℚ). The revised statement will read: if γ is eventually linearly recurrent with values and coefficients in a finitely generated ring and L_γ(z) is rational, then γ is finitely supported. This matches the scope of the specialization argument. revision: yes

Circularity Check

0 steps flagged

No circularity; algebraic reduction to finite fields is independent of the claim

full rationale

The paper proves that an eventually linearly recurrent sequence with rational Lambert series must be finitely supported by specializing coefficients and values at a finite place of a finitely generated ring, reducing to a finite field where recurrences are eventually periodic, then deriving a contradiction with rationality of the divisor-sum generating function. This chain relies on standard facts about linear recurrences over finite fields and generating functions; it does not redefine the conclusion in terms of the inputs, rename a known result, or load the argument on self-citations. The abstract omits the ring hypothesis used in the proof, but this is a statement gap rather than a circular reduction. The derivation remains self-contained against external algebraic benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The proof rests on standard algebraic facts about linear recurrences and generating functions; no free parameters or new entities are introduced.

axioms (2)

standard math Linear recurrences over a finitely generated ring become periodic after reduction at a finite place.
Invoked in the proof to obtain a contradiction with rationality unless support is finite.
standard math Rationality of the Lambert series is preserved under specialization to finite fields.
Used to transfer the rationality assumption to the finite-field setting.

pith-pipeline@v0.9.0 · 5409 in / 1142 out tokens · 45011 ms · 2026-05-07T15:25:25.815794+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

10 extracted references · 1 canonical work pages

[1]

Adamczewski and J

B. Adamczewski and J. P. Bell,The Skolem-Mahler-Lech theorem in positive characteristic and finite automata, Int. Math. Res. Not. IMRN 2019, no. 11, 3317–3352

2019
[2]

M. F. Atiyah and I. G. Macdonald,Introduction to Commutative Algebra, Addison-Wesley Publishing Co., Reading, Mass., 1969

1969
[3]

J. P. Bell,A generalised Skolem-Mahler-Lech theorem for affine varieties, J. London Math. Soc. (2)73(2006), no. 2, 367–379

2006
[4]

Eisenbud,Commutative Algebra with a View Toward Algebraic Geometry, Graduate Texts in Mathematics, vol

D. Eisenbud,Commutative Algebra with a View Toward Algebraic Geometry, Graduate Texts in Mathematics, vol. 150, Springer-Verlag, New York, 1995

1995
[5]

Everest, A

G. Everest, A. van der Poorten, I. Shparlinski, and T. Ward,Recurrence Sequences, Mathematical Surveys and Monographs, vol. 104, American Mathematical Society, Providence, RI, 2003

2003
[6]

G. H. Hardy and E. M. Wright,An Introduction to the Theory of Numbers, 6th ed., edited by R. Heath-Brown, J. Silverman, and A. Wiles, Oxford University Press, Oxford, 2008

2008
[7]

Lech,A note on recurring series, Ark

C. Lech,A note on recurring series, Ark. Mat.2(1953), 417–421

1953
[8]

A. J. van der Poorten and H. P. Schlickewei,Zeros of recurrence sequences, Bull. Austral. Math. Soc.44 (1991), no. 2, 215–223. 10 IGOR RIVIN

1991
[9]

Rivin,Walks on Free Groups and other Stories – twelve years later, arXiv:1106.5947, 2011; published as Growth in free groups (and other stories)–twelve years later,Illinois J

I. Rivin,Walks on Free Groups and other Stories – twelve years later, arXiv:1106.5947, 2011; published as Growth in free groups (and other stories)–twelve years later,Illinois J. Math.54(2010), no. 1, 327–370

work page arXiv 2011
[10]

M. D. Schmidt,A catalog of interesting and useful Lambert series identities,Online J. Anal. Comb.21(2026), Paper No. 1, 1–22. Mathematics Department, Temple University and Dimension Reducers, LLC Email address:igor@dimensionreducers.com

2026

[1] [1]

Adamczewski and J

B. Adamczewski and J. P. Bell,The Skolem-Mahler-Lech theorem in positive characteristic and finite automata, Int. Math. Res. Not. IMRN 2019, no. 11, 3317–3352

2019

[2] [2]

M. F. Atiyah and I. G. Macdonald,Introduction to Commutative Algebra, Addison-Wesley Publishing Co., Reading, Mass., 1969

1969

[3] [3]

J. P. Bell,A generalised Skolem-Mahler-Lech theorem for affine varieties, J. London Math. Soc. (2)73(2006), no. 2, 367–379

2006

[4] [4]

Eisenbud,Commutative Algebra with a View Toward Algebraic Geometry, Graduate Texts in Mathematics, vol

D. Eisenbud,Commutative Algebra with a View Toward Algebraic Geometry, Graduate Texts in Mathematics, vol. 150, Springer-Verlag, New York, 1995

1995

[5] [5]

Everest, A

G. Everest, A. van der Poorten, I. Shparlinski, and T. Ward,Recurrence Sequences, Mathematical Surveys and Monographs, vol. 104, American Mathematical Society, Providence, RI, 2003

2003

[6] [6]

G. H. Hardy and E. M. Wright,An Introduction to the Theory of Numbers, 6th ed., edited by R. Heath-Brown, J. Silverman, and A. Wiles, Oxford University Press, Oxford, 2008

2008

[7] [7]

Lech,A note on recurring series, Ark

C. Lech,A note on recurring series, Ark. Mat.2(1953), 417–421

1953

[8] [8]

A. J. van der Poorten and H. P. Schlickewei,Zeros of recurrence sequences, Bull. Austral. Math. Soc.44 (1991), no. 2, 215–223. 10 IGOR RIVIN

1991

[9] [9]

Rivin,Walks on Free Groups and other Stories – twelve years later, arXiv:1106.5947, 2011; published as Growth in free groups (and other stories)–twelve years later,Illinois J

I. Rivin,Walks on Free Groups and other Stories – twelve years later, arXiv:1106.5947, 2011; published as Growth in free groups (and other stories)–twelve years later,Illinois J. Math.54(2010), no. 1, 327–370

work page arXiv 2011

[10] [10]

M. D. Schmidt,A catalog of interesting and useful Lambert series identities,Online J. Anal. Comb.21(2026), Paper No. 1, 1–22. Mathematics Department, Temple University and Dimension Reducers, LLC Email address:igor@dimensionreducers.com

2026