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Periodic vanishings of the Legendre-17 signed partition numbers

This paper derives Rademacher-style series formulae for Legendre-signed partition numbers and utilizes these formulas alongside properties of Dedekind sums to prove that the Legendre-17 signed partition numbers vanish periodically on specific arithmetic progressions modulo 34.

Original authors: Taylor Daniels

Published 2026-03-04

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CC BY 4.0

Original authors: Taylor Daniels

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 giant bag of Lego bricks. Your job is to build towers using these bricks, but there's a catch: you can only use bricks that add up to a specific total weight, say 100 grams. You can use one big brick, or ten small ones, or any combination, as long as the total weight is exactly 100.

In mathematics, this is called partitioning a number. The number of ways you can build that tower is a famous sequence called the partition numbers.

Now, imagine we put a twist on this game. Some of our Lego bricks are "good" (they add +1 to your score), some are "bad" (they add -1), and some are "neutral" (they add 0). We want to know: if we build every possible tower of weight 100, and we add up the scores of all those towers, what is the final total?

This paper is about a very specific, tricky version of this game involving a number called 17.

The Cast of Characters

The Number 17: The paper focuses on the prime number 17. In the world of numbers, 17 has a special property related to "squares" (numbers like 1, 4, 9, 16, which are $1^2, 2^2, 3^2, 4^2$ ).
The Legendre Symbol (The Judge): This is a mathematical tool that acts like a judge. For any number, it asks: "Is this a perfect square modulo 17?"
- If yes, the judge gives a +1.
- If no, the judge gives a -1.
- If the number is a multiple of 17, the judge gives a 0.
The Signed Partition Numbers: This is the final score of our Lego game. We take every possible way to build a tower of weight $n$ , multiply the "judges' scores" of the bricks used, and sum them all up.

The Big Discovery: The Magic Vanishing Act

For most numbers, if you play this game, the final score grows and grows as the weight ( $n$ ) gets bigger. It's like a snowball rolling down a hill, getting huge.

However, the author, Taylor Daniels, discovered something magical and surprising about the number 17.

He found that for certain specific weights, the final score doesn't just get small; it becomes exactly zero. It's as if the "good" towers and the "bad" towers perfectly cancel each other out, leaving nothing behind.

It's like balancing a scale. If you have 1,000,000 heavy weights on the left and 1,000,000 heavy weights on the right, the scale doesn't tip; it stays perfectly flat.

The Specific Rules for Vanishing:
The paper proves that the score is zero if the weight $n$ follows these patterns:

Scenario A: If $n$ $n$ is an odd number, and when you do a specific math trick ( $1 - 24n$ $1 - 24 n$ ), the result is a "fourth-power square" modulo 17.
- Result: The score is 0 for weights like 17, 19, 25, 27, 51, 53, etc.
Scenario B: If $n$ $n$ is an odd number, and the math trick result is a "non-square" modulo 17.
- Result: The score is 0 for weights like 11, 15, 29, 33, 45, 49, etc.

How Did They Prove It? (The Detective Work)

You can't just check every single number up to infinity; there are too many. So, the author used a sophisticated mathematical toolkit to prove this happens forever.

The Rademacher Formula (The Blueprint): He used a complex mathematical recipe (a series formula) that breaks the problem down into smaller, manageable pieces. Think of it like taking apart a giant clock to see how the gears turn.
The "Kloosterman Sums" (The Hidden Gears): Inside the recipe, there are specific parts called "sums." The author showed that for the specific numbers where the score should be zero, these hidden gears stop moving entirely. They cancel each other out perfectly.
Dedekind Sums (The Balance Scale): He used properties of "Dedekind sums" (a type of mathematical balance) to show that the "good" and "bad" contributions are perfectly symmetrical for these specific numbers.

The "Only Us" Club

The author also did some detective work to see if this happens with other numbers.

He checked the number 5. It also has this "vanishing" magic (it was known before).
He checked thousands of other numbers up to 2,000.
The Conclusion: It seems that only 5 and 17 have this special "perfect cancellation" property. They are the only two numbers in the universe of primes that make the Lego tower scores disappear completely on specific patterns.

Why Does This Matter?

In the world of math, finding a pattern where things cancel out to zero is like finding a hidden door in a maze. It suggests a deep, underlying symmetry in how numbers interact.

This paper doesn't just say "it happens"; it builds a bridge from the messy, chaotic world of counting partitions to the clean, orderly world of symmetry and algebra. It tells us that even in the seemingly random world of adding numbers up, there are hidden rules that make things disappear into thin air, and for the number 17, those rules are incredibly precise.

In short: The paper proves that for the number 17, there are specific "magic numbers" where the sum of all possible weighted combinations is exactly zero, and it explains why using a beautiful, complex dance of mathematical formulas.

1. Problem Statement

The paper investigates the behavior of signed partition numbers, denoted $p(n, f)$ , where $f: \mathbb{N} \to \{0, \pm 1\}$ is a weighting function. Specifically, the author focuses on the case where $f$ is the Legendre symbol $\chi = \left(\frac{\cdot}{p}\right)$ modulo a prime $p$ .
The quantity is defined as:
$p(n, \chi) = \sum_{\substack{x_1 + \dots + x_k = n \\ x_1 \ge \dots \ge x_k > 0}} \chi(x_1)\chi(x_2)\dots\chi(x_k)$
While it is known that for most primes $p$ , $p(n, \chi)$ grows asymptotically (similar to the ordinary partition function $p(n)$ ), previous work established that for $p=5$ , the signed partition numbers vanish periodically (e.g., $p(n, (\frac{\cdot}{5})) = 0$ for $n \equiv 2 \pmod{10}$ ).

The Core Question: Are there other primes $p$ (specifically those where $p \equiv 1 \pmod 8$ , a condition where growth is usually expected) for which $p(n, \pm \chi)$ vanishes on specific arithmetic progressions?

2. Methodology

The paper employs a rigorous analytic number theory approach, combining classical partition theory with advanced properties of modular forms and Dedekind sums.

A. Rademacher-Style Series Expansion

The author derives exact infinite series formulae for $p(n, \chi)$ and $p(n, -\chi)$ by adapting the Rademacher circle method and techniques from Lehner and Hagis Jr.

Generating Functions: The generating function $\Phi(x) = \prod_{n \ge 1} (1 - \chi(n)x^n)^{-1}$ is analyzed.
Functional Equations: The paper derives functional equations for $\Phi(x)$ near the unit circle, categorized by the greatest common divisor $\gcd(k, 2p)$ of the denominator $k$ in the Farey arc expansion.
Series Decomposition: The partition function is decomposed into four components based on $\gcd(k, 2p) \in \{1, 2, p, 2p\}$ . The terms corresponding to $\gcd(k, 2p) = 2p$ are shown to vanish identically.
Bessel Functions: The remaining terms are expressed as sums involving modified Bessel functions of the first kind, $I_1(z)$ , weighted by Kloosterman-type sums $L_k(n)$ and $L^+_k(n, m)$ .

B. Analysis of Coefficients (Kloosterman Sums)

The crux of the proof lies in analyzing the coefficients $L_k(n)$ and $L^+_k(n, m)$ , which are exponential sums involving the root of unity $\phi_{h,k} = \exp(\pi i \Lambda(h, k))$ .

Dedekind Sums: The exponent $\Lambda(h, k)$ is expressed in terms of standard Dedekind sums $s(h, k)$ and "character-twisted" Dedekind sums $s_\chi(h, k)$ .
Congruence Analysis: The author establishes deep congruences for these sums modulo 16 and modulo $3K$ $3 K$ (where $K=kp$).
- Key tools include reciprocity laws for $s_\chi$ , properties of periodic Bernoulli numbers $B_2(\chi)$ , and combinatorial counting of residues (quantities denoted $\tau_{es}$ and $\tau_{er}$ ).
Cancellation Mechanism: The proof demonstrates that for specific residue classes of $n$ , the terms in the series cancel each other out perfectly due to symmetries in the exponential sums. Specifically, for every term with a "quartic" residue $h_1$ , there exists a corresponding "quadratic-nonquartic" residue $h_2$ such that their contributions sum to zero.

3. Key Contributions and Results

A. Main Theorems

The paper rigorously proves that for the prime $p=17$ , the signed partition numbers vanish on specific arithmetic progressions modulo 34.

Theorem 1.1:
$p\left(n, \left(\frac{\cdot}{17}\right)\right) = 0 \quad \text{for all } n \equiv 17, 19, 25, 27 \pmod{34}.$
Equivalently, $p(n, \chi) = 0$ if $n$ is odd and $1-24n$ is a quartic residue modulo 17.
Theorem 1.2:
$p\left(n, -\left(\frac{\cdot}{17}\right)\right) = 0 \quad \text{for all } n \equiv 11, 15, 29, 33 \pmod{34}.$
Equivalently, $p(n, -\chi) = 0$ if $n$ is odd and $1-24n$ is a quadratic-nonquartic residue modulo 17.

B. Specific Formulae for $p=17$

The author provides explicit Rademacher-style series for $p=17$ , calculating specific constants such as:

$B_2(\chi) = 8$ for $p=17$ .
The constants $c_m$ in the series expansion, which restrict the summation to $m \in \{0, 1, 2\}$ .
The explicit values of the coefficients $\sigma^+_m$ for the auxiliary generating functions.

C. Conjecture on Uniqueness

Based on numerical experiments for $n \le 10,000$ and primes $p \le 2000$ , the author conjectures that $p=5$ and $p=17$ are the only primes for which $p(n, \pm \chi)$ exhibits periodic vanishing on arithmetic progressions modulo $2p$ .

4. Significance

Extension of Partition Theory: This work extends the known phenomenon of periodic vanishing (previously only observed for $p=5$ ) to a new prime $p=17$ . It challenges the intuition that Legendre-signed partitions generally grow exponentially for $p \equiv 1 \pmod 8$ .
Analytic Techniques: The paper provides a sophisticated application of the Rademacher circle method to signed partitions, successfully handling the complex "character-twisted" Dedekind sums that arise from the Legendre symbol.
Connection to Quartic Residues: The results reveal a deep connection between the vanishing of partition numbers and the arithmetic properties of quartic residues modulo 17. The vanishing condition is directly linked to whether $1-24n$ is a quartic residue.
Foundation for Future Research: The derived series formulae and the detailed analysis of the coefficients $L_k$ and $L^+_k$ provide a framework that could be applied to investigate other primes or different character sums, although the author notes the notation becomes increasingly complex for general $p$ .

In summary, Taylor Daniels provides a rigorous proof that the Legendre-signed partition numbers for $p=17$ vanish periodically, utilizing a blend of classical analytic number theory and novel congruence properties of twisted Dedekind sums.