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Linear recurrences and rational Lambert series

This paper establishes that for a sequence with an eventually linearly recurrent ordinary generating function, its associated Lambert series is rational if and only if the sequence is finitely supported, a result proven by leveraging the periodicity of recurrences over finite fields.

Original authors: Igor Rivin

Published 2026-04-29

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: Igor Rivin

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you have a long, endless string of beads, where each bead is a number. In mathematics, this string is called a sequence. Sometimes, these sequences follow a strict rule: to get the next number, you just add up the previous few numbers in a specific way. Mathematicians call this a "linear recurrence." If you write down these numbers as a giant formula (a "generating function"), the formula is usually a simple fraction (a "rational function").

This paper is about what happens when you look for zeros in these strings of numbers. Specifically, it asks: What does it mean if a sequence has a lot of zeros? And what happens if you mix these sequences together in a special way called a "Lambert series"?

Here is the breakdown of the paper's discoveries, using simple analogies.

1. The "Zero Pattern" Rule (The Skolem-Mahler-Lech Theorem)

The paper starts with a known rule about sequences that follow a strict pattern. If you look at all the places where the number is zero, they don't appear randomly. They appear in a very predictable way.

The Analogy: Imagine a train schedule. The train stops at a station (a zero) at 1:00, 1:05, 1:10, and so on. Or maybe it stops at 1:00, 1:03, 1:06.
The Finding: The paper confirms that if a sequence has zeros, those zeros eventually fall into a pattern of "arithmetic progressions" (like every 5th number, or every 7th number). It's like a train that eventually settles into a strict timetable.

2. The "Prime Number" Test

The authors looked at a specific scenario: What if the numbers at all the prime number positions (2, 3, 5, 7, 11...) are zero?

The Finding: If the numbers at all prime spots are zero, the whole sequence isn't just random; it's actually made of smaller, simpler sequences stacked on top of each other.
The Analogy: Imagine a complex musical chord. If you notice that the notes played on every "prime beat" are silent, you realize the music isn't one big messy jam. Instead, it's actually three separate, simpler melodies playing at different speeds (like one melody every 2 beats, another every 3 beats). The paper proves you can break the complex formula down into these simpler, "proper power" parts.

3. The "Root of Unity" Connection

The paper also explains why a sequence might have infinitely many zeros. It turns out this happens only if the "engine" driving the sequence has a specific symmetry.

The Analogy: Think of the sequence as a spinning wheel. If the wheel has a "root of unity" relation, it's like the wheel has a gear that clicks back to the start after a few turns. If the gears don't line up perfectly (no root-of-unity relation), the wheel spins forever without hitting the same spot twice, and you won't get a repeating pattern of zeros.
The Finding: If you see infinitely many zeros, the "gears" (the mathematical poles of the function) must be related in a way that allows them to line up and cancel each other out periodically.

4. The Main Event: The Lambert Series Rigidity

This is the paper's biggest discovery. A Lambert series is a special way of mixing a sequence. Instead of just listing the numbers, you take each number and spread it out over all its multiples.

Formula: If your sequence is $\gamma$ , the Lambert series adds up $\gamma_1$ , then $\gamma_1 + \gamma_2$ , then $\gamma_1 + \gamma_2 + \gamma_3$ , and so on, but weighted by how they divide into numbers.

The Big Question: If you start with a sequence that follows a strict rule (linear recurrence), and you turn it into a Lambert series, and the result also follows a strict rule (is rational), what does that tell you about the original sequence?

The Answer: The original sequence must be finite.

The Analogy: Imagine you have a machine that takes a stream of water (your sequence) and sprays it into a giant, complex fog (the Lambert series).
- If the water stream is endless and follows a pattern (like a river), the fog will be chaotic and messy. It won't form a simple shape.
- The only way for the fog to form a simple, clean shape (a rational function) is if the water stream stops after a while.
The Conclusion: If both the original sequence and its Lambert series are "nice" and follow simple rules, the original sequence must have been short and finite to begin with. It cannot be an endless, repeating pattern.

5. How They Proved It (The "Finite Field" Trick)

The authors didn't just guess this; they used a clever mathematical trick to prove it.

The Method: They took the complex numbers involved in the sequence and "reduced" them, like taking a high-resolution photo and shrinking it down to a tiny, low-resolution grid (a finite field).
The Logic: In this tiny, low-resolution world, patterns become very simple and repeat quickly (they become periodic). They showed that if the original sequence were infinite and non-zero, this tiny world would produce a mathematical contradiction (like saying $1 = 0$ ).
The Result: Because the tiny world breaks if the sequence is infinite, the sequence must be finite.

Summary of Examples

The paper uses this logic to prove some fun facts:

Fibonacci Numbers: The famous Fibonacci sequence (1, 1, 2, 3, 5, 8...) is endless and follows a rule. Therefore, if you make a Lambert series out of it, the result is not a simple rational function. It's too messy.
Periodic Sequences: If you have a sequence that just repeats forever (like 1, 2, 1, 2...), its Lambert series is only "nice" if the sequence is actually all zeros.

In a nutshell: This paper proves that you can't take a long, endless, patterned sequence, mix it into a Lambert series, and get a simple result back. If the result is simple, the input must have been short and finite. It's a "rigidity" theorem: the structure is so tight that it forces the sequence to stop.

1. Problem Statement and Context

The paper investigates the structural constraints on sequences and their generating functions, specifically focusing on the interplay between linear recurrence sequences, rational power series, and Lambert series.

Core Objects:
- Rational Power Series: A series $F(z) = \sum a_n z^n$ is rational if and only if its coefficients $(a_n)$ satisfy a constant-coefficient linear recurrence relation (eventually).
- Lambert Series: Given a sequence $\gamma = (\gamma_n)_{n \ge 1}$ , the Lambert series is defined as $L_\gamma(z) = \sum_{n \ge 1} \frac{\gamma_n z^n}{1-z^n}$ . The coefficients of $L_\gamma(z)$ are the divisor sums of $\gamma$ : $b_m = \sum_{d|m} \gamma_d$ .
The Central Question: If a sequence $\gamma$ is "well-behaved" (specifically, eventually linearly recurrent) and its associated Lambert series $L_\gamma(z)$ is also "well-behaved" (rational), what does this imply about the structure of $\gamma$ ?
Motivation: The work stems from rigidity problems in free groups and the behavior of the operator $T_\gamma G(z) = \sum \gamma_n G(z^n)$ .

2. Methodology

The paper employs a blend of classical analytic number theory, algebraic geometry, and $p$ -adic analysis. The methodology is divided into two main phases:

A. Analysis of Rational Power Series (Skolem-Mahler-Lech Framework)

The author utilizes the Skolem-Mahler-Lech (SML) theorem, which states that the zero set of a linear recurrence sequence is a finite union of arithmetic progressions plus a finite set.

Prime-Index Vanishing: The paper analyzes the case where coefficients at prime indices vanish ( $a_p = 0$ for large $p$ ). Using Dirichlet's theorem on primes in arithmetic progressions, it deduces that the support of the sequence must be restricted to residue classes not coprime to the period. This leads to a decomposition of the rational function into sums of functions of proper powers ( $z^d$ ).
Root-of-Unity Relations: It establishes that if a rational power series has infinitely many zero coefficients, the poles of minimum modulus must satisfy a root-of-unity relation (i.e., the ratio of two dominant poles is a root of unity).

B. Finite-Place Reduction (The Lambert Rigidity Proof)

The proof of the main theorem regarding Lambert series avoids the heavy machinery of $p$ -adic interpolation used in the full SML theorem. Instead, it uses a finite-field reduction strategy:

Algebraic Setup: The recurrence coefficients and sequence values are contained in a finitely generated domain $A$ over $\mathbb{Z}$ .
Finite Specialization: Using a lemma on finitely generated algebras, the author constructs a homomorphism $\phi$ from this domain to a finite field $\mathbb{F}_q$ . This preserves non-zero elements (specifically a key divisor sum $S$ ) while mapping the recurrence relations to the finite field.
Periodicity in Finite Fields: Over a finite field, any linear recurrence sequence is eventually periodic.
The Contradiction: The proof compares the values of the divisor sum sequence $b_n$ $b_{n}$ at indices $mq$ and $mq^2$ $m q^{2}$ (where $q$ $q$ is a large prime).
- Due to periodicity, $b_{mq^2} \equiv b_{mq} \pmod{\mathbb{F}_q}$ .
- However, algebraic manipulation of the divisor sum definition shows that $b_{mq^2} - b_{mq} = \sum_{e|m} \gamma_{eq^2}$ .
- Because $\gamma$ is periodic modulo $\mathbb{F}_q$ , $\gamma_{eq^2} \equiv \gamma_e$ .
- This implies the difference equals the original divisor sum $S = \sum_{e|m} \gamma_e$ .
- Since $S$ was chosen to be non-zero and invertible in the finite field, this yields a contradiction ( $S \equiv 0$ vs $S \neq 0$ ).

3. Key Contributions and Results

Main Theorem (Theorem 1.1)

Statement: Let $\gamma = (\gamma_n)$ be a sequence of complex numbers. If both $\gamma$ and its divisor sum sequence $b_n = \sum_{d|n} \gamma_d$ are eventually linearly recurrent, then $\gamma$ is finitely supported (i.e., $\gamma_n = 0$ for all sufficiently large $n$ ).

Corollary 1.2 (Generating Function Form):
If the ordinary generating function $\Gamma_\gamma(z)$ is rational and the Lambert series $L_\gamma(z)$ is rational, then $\Gamma_\gamma(z)$ must be a polynomial.

Significance: This establishes a "rigidity" for Lambert series. Among sequences with rational generating functions, only the trivial (finitely supported) ones produce rational Lambert series.

Structural Consequences for Rational Functions

Proposition 3.1: If a rational function $F(z)$ has zero coefficients at all sufficiently large prime indices, then $F(z)$ can be decomposed into a polynomial plus a finite sum of rational functions in proper powers $z^d$ (where $d > 1$ ).
Proposition 3.3: If a rational power series has infinitely many zero coefficients, the poles of minimum modulus must include at least two whose ratio is a root of unity.

Specific Examples and Applications

Fibonacci Sequence: The Lambert series of the Fibonacci numbers is not rational (Corollary 6.5).
Periodic Coefficients: If $\gamma$ is periodic and $L_\gamma(z)$ is rational, then $\gamma$ must be identically zero (Corollary 6.4).
Prime vs. Prime-Square: Proposition 6.1 shows that for such sequences, $\gamma_p = 0$ and $b_p = \gamma_1$ for all sufficiently large primes $p$ .

4. Significance and Novelty

Rigidity of Lambert Series: The paper resolves a fundamental question about the operator $T_\gamma$ . It proves that the "Lambert transform" destroys rationality unless the input sequence is trivial. This contrasts with the ordinary generating function, where rationality is preserved under many operations.
Simplification of SML Techniques: While the Skolem-Mahler-Lech theorem typically requires $p$ -adic analytic interpolation (Strassmann's theorem) to prove that a subsequence vanishes identically, this paper demonstrates that for the specific structure of Lambert series, a finite-field reduction is sufficient to generate a contradiction. The "divisor-sum identity" $b_{mq^2} - b_{mq} = S$ provides the necessary non-vanishing quantity immediately, bypassing the need for $p$ -adic analysis.
Interplay of Number Theory and Algebra: The work elegantly connects the distribution of primes (Dirichlet's theorem), the structure of recurrence relations, and the arithmetic of divisor sums. It highlights how the multiplicative structure of the integers (via the divisor sum) imposes severe constraints on additive structures (linear recurrences).

Conclusion

Igor Rivin's paper establishes a strong rigidity theorem: the only sequences that are both linearly recurrent and possess a rational Lambert series are those with finite support. The proof innovates by adapting the finite-place reduction technique of the Skolem-Mahler-Lech theorem but leveraging the specific algebraic properties of divisor sums to achieve a contradiction without deep $p$ -adic machinery. This result has immediate implications for the study of generating functions in combinatorics and number theory, ruling out the rationality of Lambert series for many classical sequences (like Fibonacci or periodic sequences).