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Double $L$ -values of Negative Weight Admitting Series Representations

This paper constructs double $L$ -values of negative weight that admit series representations by utilizing the Thue-Morse sequence from combinatorics on words and automatic sequences.

Original authors: Daiki Iju, Takashi Nakamura

Published 2026-07-17

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Daiki Iju, Takashi Nakamura

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 a vast, infinite library where every book is a number, and the stories written inside them are patterns that repeat forever. In the world of mathematics, there is a special corner dedicated to "L-values." Think of these as the secret codes hidden inside infinite sums—long lists of fractions added together that, if they behave nicely, settle down to a single, specific number. For a long time, mathematicians believed these codes only worked when the numbers in the denominators were "heavy" enough (positive weights) to keep the sum from exploding into infinity. It was like trying to build a tower: if the bricks were too light or negative, the whole thing was thought to collapse. But recently, a pair of researchers discovered that with the right blueprint, you can actually build a stable tower even with "negative" bricks, provided you arrange them with a very specific, rhythmic pattern.

This paper, written by Daiki Iju and Takashi Nakamura, dives into this surprising possibility. They are working with "double L-values," which are like two-story towers of these infinite sums. The authors tackle a tricky problem: what happens when the weight of the sum is negative? Usually, this leads to chaos. However, they prove that by using a special sequence of numbers known as the "Thue-Morse sequence"—a pattern that looks like a coin flip but is actually determined by the number of ones in a number's binary code—you can construct a series that converges. In other words, they found a way to make an infinite sum of negative-weight terms settle down to a finite, meaningful answer, effectively rewriting the rules of how these mathematical structures can behave.

The Story of the Infinite Sum

To understand what Iju and Nakamura achieved, let's first look at the "rules of the game" they are playing. In the world of multiple L-values, mathematicians add up fractions in a very specific order. Imagine you have a list of numbers, and you are adding up terms like $1/m_1^{k_1} \times 1/m_2^{k_2}$ , where $m_1$ is bigger than $m_2$ , and so on. The "weight" of this sum is the total of the exponents ( $k_1 + k_2$ ).

For a long time, the rule of thumb was simple: if the weight is positive and big enough (specifically, if the first exponent is 2 or more), the sum settles down nicely. But if the weight is negative, the terms get bigger and bigger as you go further out, and the sum was thought to blow up to infinity. In 2004, a famous mathematician named Arakawa suggested a condition for when these sums might still work even if the first exponent was 1, but a later study showed that even that condition wasn't the whole story.

This is where our authors step in. They ask a bold question: Can we make a sum with a negative weight (like -1, -2, etc.) actually converge? The answer, they prove, is a resounding "yes," but only if we are very clever about how we choose the numbers in the numerator.

The Magic of the Thue-Morse Sequence

The secret weapon in their construction is something called the Thue-Morse sequence. You can think of this sequence as a cosmic rhythm. To generate it, you look at a number in binary (base 2, using only 0s and 1s). If the number of 1s is even, the sequence gives you a 0; if it's odd, it gives you a 1.

0 (zero 1s) $\rightarrow$ 0
1 (one 1) $\rightarrow$ 1
2 (one 1) $\rightarrow$ 1
3 (two 1s) $\rightarrow$ 0
4 (one 1) $\rightarrow$ 1
...and so on.

This pattern, which appears in everything from the growth of plants to the design of antennas, has a magical property: it balances itself out perfectly over large ranges. This balancing act is the key to taming the wild, negative-weight sums.

The Construction: Building a Stable Tower

The authors construct two specific functions, $f_1$ and $f_2$ , which act as the "ingredients" for their infinite sum.

The First Ingredient ( $f_1$ ): This is a simple, repeating pattern that is non-zero only at specific intervals. It sets the stage.
The Second Ingredient ( $f_2$ ): This is the complex part. It is built by mixing the Thue-Morse sequence with some fancy mathematical waves (roots of unity). The authors carefully tune this mixture so that the "positive" and "negative" parts of the sum cancel each other out almost perfectly, leaving just a tiny, manageable remainder.

They prove that when you combine these ingredients into a "double L-value" with a negative weight (specifically $-l$ ) and a second weight of 2, the infinite series converges. This means that even though the individual terms might be getting larger, the specific way they are arranged causes them to cancel out in a way that the total sum stays finite.

Why This Matters

The most exciting part of their discovery is that the sequence $f_2$ they created is not just a one-off trick. It is non-zero at infinitely many points, meaning the pattern is rich and complex, not just a simple zeroing-out. They show that for any non-negative integer $l$ , you can build such a convergent series.

This result is significant because it shatters the old intuition that negative weights always lead to divergence. It shows that with the right combinatorial "dance" (using the Thue-Morse sequence), you can stabilize even the most unruly mathematical sums. The authors didn't just guess this; they provided a rigorous proof, showing exactly how the terms cancel out and estimating the size of the remaining error to prove it vanishes as the sum goes to infinity.

In short, Iju and Nakamura have opened a new door in the library of numbers. They showed us that even in the realm of negative weights, where chaos was expected, there is a hidden order waiting to be discovered, provided you know the right rhythm to play.

Technical Summary: Double L-Values of Negative Weight Admitting Series Representations

Problem Statement
The paper addresses the convergence properties of multiple $L$ -values, specifically focusing on the case of negative weights. In the theory of multiple zeta and $L$ -values, series of the form $L^*(k_1, \dots, k_n; f_1, \dots, f_n)$ are typically defined for positive integer weights $k_i$ . When the first weight $k_1 = 1$ , convergence is generally contingent upon the condition $\sum_{y \in R} f_1(y) = 0$ . A previous result by Arakawa and Kaneko suggested that if this sum is non-zero, the series diverges. However, a counter-example provided by Ageji, Iju, and Nakamura demonstrated that convergence is possible even when the sum is non-zero, provided the weight $k_1$ is an integer $j \le 1$ and specific conditions on subsequent weights and functions are met.

The central problem this paper tackles is whether convergent multiple $L$ -values can be constructed specifically for negative weights ( $k_1 = -l$ where $l \in \mathbb{N}_{\ge 0}$ ), a regime where standard absolute convergence fails and the series are not naturally defined.

Methodology
The authors construct specific functions $f_1$ and $f_2$ within the space of mappings from a finite ring $R_{2l+3}$ to a cyclotomic field. The construction relies heavily on the Thue–Morse sequence ( $t_j$ ), a binary sequence defined by the parity of the number of 1s in the binary expansion of $j$ .

Function Construction:
- $f_1$ is defined as a periodic function related to the roots of unity, specifically designed such that its sum over the period is non-zero.
- $f_2$ is constructed as a linear combination of auxiliary functions $g_{j1}, g_{j2},$ and $h_j$ , weighted by the Thue–Morse sequence terms $(-1)^{t_j}$ . These auxiliary functions are defined via finite Fourier expansions involving roots of unity $\xi_{2l+2}$ and $\xi_{2l+3}$ .
Utilization of the Prouhet–Tarry–Escott Problem:
The proof leverages a classical result (Theorem C) regarding the Thue–Morse sequence, which states that for any $l$ , the sums of powers of elements in the sets defined by the sequence (where $t_j=1$ vs $t_j=0$ ) are equal up to degree $l-1$ . This property is used to cancel out lower-order terms in the asymptotic expansion of the series.
Asymptotic Analysis:
To prove convergence, the authors analyze the partial sums of the series $L^*(-l, 2; f_1, f_2)$ . They decompose the tail of the series into components ( $H(N), G_1(N), G_2(N)$ ) and apply the binomial theorem to expand the denominators. By utilizing the vanishing power sums from the Prouhet–Tarry–Escott property, they show that the leading terms in the expansion cancel out, leaving a remainder term that decays sufficiently fast ( $O(N^{-(l+2)})$ ) to ensure the convergence of the double sum.

Key Contributions and Results
The primary result is Theorem 1.1, which states:

For any $l \in \mathbb{N}_{\ge 0}$ , there exist functions $f_1, f_2 \in F(R_{2l+3}; \mathbb{Q}(\xi_{2l+2}))$ such that the series $L^*(-l, 2; f_1, f_2)$ converges.

Key technical findings supporting this theorem include:

Non-triviality: The constructed sequence $f_2$ is not identically zero; it takes non-zero values at infinitely many integers (Lemma 3.1).
Convergence Mechanism: The convergence is achieved not by the absolute convergence of the individual terms (which fails for negative weights), but by the conditional convergence resulting from the specific cancellation properties induced by the Thue–Morse sequence and the Prouhet–Tarry–Escott relations.
Explicit Construction: The paper provides explicit formulas for $f_1$ and $f_2$ in terms of roots of unity and the Thue–Morse sequence, rather than relying on existence proofs alone.

Significance
The paper claims significance by extending the known domain of convergent multiple $L$ -values to include negative weights. While previous work had addressed the case of $k_1=1$ with non-vanishing sums, this work demonstrates that convergence is achievable even when the weight is negative ( $k_1 = -l$ ). This is accomplished by carefully engineering the periodic functions $f_1$ and $f_2$ to exploit the combinatorial properties of the Thue–Morse sequence, thereby creating a series representation that converges despite the negative exponent in the denominator. The result challenges the intuition that negative weights inherently preclude the existence of convergent series representations in this context.

The Story of the Infinite Sum

The Magic of the Thue-Morse Sequence

The Construction: Building a Stable Tower

Why This Matters

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