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Anomalous Partial Quotients in the Continued Fraction of $\sqrt{ζ(3)-S_N}$

This paper proves that for specific indices $N_k$ derived from companion Pell numbers, the continued fraction of $\sqrt{\zeta(3)-S_{N_k}}$ exhibits anomalous partial quotients, including a cubic growth term, by combining Euler--Maclaurin expansions with Pell identities and verifying the results using the LEAN proof assistant.

Original authors: David Victor Feldman

Published 2026-07-07

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CC BY 4.0

Original authors: David Victor Feldman

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 are trying to measure the length of a very strange, invisible rope. This rope represents a tiny gap between a famous mathematical constant (called $\zeta(3)$ ) and a specific sum of numbers. Usually, when mathematicians try to measure this gap using a special tool called a "continued fraction," the numbers they get are predictable and grow in a steady, boring way—like a staircase where every step is roughly the same size.

However, this paper discovers a secret shortcut. At very specific, rare locations along the number line (called "Pell indices"), the behavior of this rope changes dramatically. It's as if you were walking up a normal staircase, and suddenly, every third step becomes a giant leap, while the steps in between become tiny, almost invisible hops.

Here is a breakdown of what the paper actually found, using simple analogies:

1. The "Normal" Behavior vs. The "Anomaly"

Think of the continued fraction as a recipe for breaking a number down into a sequence of whole numbers (partial quotients).

The Generic Rule: For most numbers, these recipe steps grow slowly and linearly. If you go from step 1 to step 100, the numbers get a little bigger, but not by much.
The Anomaly: The author, David Feldman, found that at specific "magic numbers" (derived from a sequence called Pell numbers), the recipe goes haywire.
- The first number in the sequence is huge (linear growth).
- The second number is just "1" (tiny).
- The third number explodes! It grows cubically. If the first number is 10, the third might be 1,000 or 10,000. It's like a staircase where one step is a skyscraper.
- The fourth number is "1" again.
- The fifth and sixth numbers follow a complex, repeating pattern based on remainders.

2. The "Magic Numbers" (The Pell Connection)

Why does this happen only at specific spots? The paper links this to a special family of numbers called Pell numbers (0, 1, 2, 5, 12, 29...).

Imagine these numbers are keys to a hidden door. When you plug a Pell number into the formula for our "invisible rope," the math aligns perfectly.
The paper proves that at these specific keys, the irrational parts of the math (the messy square roots) cancel out in a very precise way, leaving behind a clean, predictable pattern that reveals the "giant leap" (the cubic growth).

3. How They Proved It (The Two-Step Dance)

The author didn't just guess this pattern; he proved it using two different methods, like checking a map with two different compasses:

Method A: The High-Precision Telescope (Euler-Maclaurin Expansion)
The author used a powerful mathematical telescope to zoom in on the "gap" with extreme precision. He expanded the formula into a long series of terms. By calculating the first few terms with extreme accuracy, he could see exactly where the "floor" (the integer part) of the numbers would land. This confirmed the giant cubic jump.
Method B: The "Rationalization" Trick
The hardest part was proving the very first number in the sequence. To do this, the author used a clever trick called "rationalization."
- Analogy: Imagine you have a wobbly, irrational number (like $\sqrt{2}$ ) multiplied by a fraction. It's hard to measure. But if you multiply the top and bottom by a specific "conjugate" (a partner number), the messy square root disappears, and you are left with a clean, rational number.
- By doing this, the author showed that the value sits strictly between two integers (specifically, between $M-1$ and $M$ ), proving exactly what the first number in the sequence must be.

4. The Computer Verification

The paper mentions that all these claims have been checked by a computer program called Lean (with help from a tool named Aristotle).

Analogy: Think of this as a super-strict referee. The author wrote down the proof, and the computer read every single logical step to ensure there were no mistakes, no skipped steps, and no "hand-waving." The computer gave a "green light" to the entire argument.

5. What About the Rest? (Open Questions)

The paper doesn't claim to solve everything. It leaves a few mysteries on the table:

The "Heuristic" Guess: The author has a strong hunch (a heuristic) about what the numbers look like before the magic Pell indices kick in, but he hasn't rigorously proved the very first few steps of that guess yet. It's like seeing a pattern in the clouds but not having the physics to prove it yet.
Beyond the Sixth Step: The paper calculates the first six numbers of the sequence perfectly. What happens at the seventh, eighth, or hundredth step? The pattern gets complicated, and the author admits that describing the full, long-term behavior is still an open problem.
Why $\zeta(3)$ ? The author explains that this specific "cubic explosion" only happens because of the specific power of the numbers involved (related to the number 3). If you changed the math slightly, the giant leap wouldn't happen.

Summary

In short, this paper is a discovery of a mathematical glitch (or feature, depending on how you look at it). It shows that for a specific, hard-to-calculate number, the usual rules of "how big the next number is" break down at specific intervals, creating a sequence with a massive, cubic-sized jump. The author proved this using high-precision math, clever algebraic tricks, and computer verification, revealing a hidden structure in the numbers that was previously invisible.

Technical Summary: Anomalous Partial Quotients in the Continued Fraction of $\sqrt{\zeta(3) - S_N}$

Problem Statement
The paper investigates the behavior of the simple continued fraction expansion of the quantity $\sqrt{R_N}$ , where $R_N = \zeta(3) - S_N$ and $S_N = \sum_{j=1}^N j^{-3}$ . While the partial quotients of $\sqrt{R_N}$ are generically of size $O(N)$ , the author identifies a specific sequence of indices $N_k$ where the expansion exhibits "anomalous" behavior. Specifically, at these indices, the third partial quotient grows cubically with respect to the associated Pell numbers, deviating significantly from the linear growth observed in generic cases.

Methodology
The proof is self-contained and proceeds in two distinct stages, combining analytic number theory with formal verification:

Euler–Maclaurin Expansion and Rationalization (Stages 1 & 2):
- The author utilizes the Euler–Maclaurin expansion to approximate $1/\sqrt{R_{N_k}}$ with high precision. A critical algebraic feature of the chosen indices $N_k$ is that they allow the cancellation of irrational terms (specifically $\sqrt{2}$ ) when the expansion is expressed in terms of Pell numbers $M_k = P_{2k+1}$ and companion Pell numbers $Q_{2k+1}$ .
- This leads to a Laurent polynomial expansion in $M_k$ with rational coefficients.
- To rigorously establish the first partial quotient ( $PQ_0$ ), the paper employs a "rationalization" argument. By multiplying the irrational term $\sqrt{R_N}$ by a rational approximation derived from the Pell identity $Q^2 - 2M^2 = -1$ , the author bounds the value within the interval $(M_k - 1, M_k)$ , proving $PQ_0 = M_k - 1$ .
Continued Fraction Chain and Heuristic Derivation:
- Using the high-precision expansion, the author performs successive "floor-and-invert" steps to derive the first six partial quotients.
- A heuristic derivation is provided using Gosper's bihomographic continued-fraction algorithm. This algorithm treats the problem as the continued fraction of a product of two numbers ( $1/\sqrt{R_N} = 1/(\sqrt{R_N/2} \cdot \sqrt{2})$ ), offering insight into why the Pell indices create a resonance that produces the anomalous cubic term.

Key Results
The paper establishes Theorem 1.1, which explicitly determines the first six partial quotients of the continued fraction for $\sqrt{R_{N_k}}$ for all $k \geq 3$ . Let $M_k = P_{2k+1}$ and $r_k = (10M_k) \pmod{261}$ . The expansion begins:
$\sqrt{R_{N_k}} = [0; M_k - 1, 1, 6M_k^3 + 12M_k - 2, 1, \lfloor \frac{10M_k - 261}{261} \rfloor, \lfloor \frac{261}{r_k} \rfloor + \epsilon_k, \dots]$
where $\epsilon_k = -1$ if $r_k$ divides 261 (occurring at specific periodic indices) and $0$ otherwise.

Growth Rates: The partial quotients exhibit distinct growth behaviors:
- $PQ_0 \sim M_k$ (Linear)
- $PQ_1 = 1$
- $PQ_2 \sim 6M_k^3$ (Cubic, the primary anomaly)
- $PQ_3 = 1$
- $PQ_4 \sim \frac{10}{261}M_k$ (Linear)
- $PQ_5$ is bounded (at most 261) and depends on the modular recurrence of $r_k$ .
Formalization: All claimed results have been formalized in the LEAN proof assistant with the aid of the Aristotle library. The formalization includes the Euler–Maclaurin bounds and the rationalization proof for the first partial quotient.

Significance and Open Problems
The paper highlights the "delicate" nature of the first step ( $PQ_0 = M_k - 1$ ), which relies on the precise cancellation of $\sqrt{2}$ terms inherent to the Pell indices. The cubic growth of $PQ_2$ is identified as the mechanism that makes these indices exceptional.

The author notes several limitations and open problems:

Heuristic vs. Rigorous: While the first six quotients are proven, a heuristic derivation for the continued fraction of the intermediate function $g(N) = \sqrt{R_N/2}$ (Conjecture 4.1) remains unproven for the smallest indices ( $k=1$ ) due to remainder bounds. This is the only unproven claim in the accompanying Lean formalization.
Generalization: The specific cubic anomaly is tied to $\zeta(3)$ (where $s=3$ ). For other odd $s$ , the leading growth rate changes, and the resonance mechanism breaks down.
Future Directions: The paper poses questions regarding the full quasi-periodic structure of partial quotients beyond $PQ_5$ , potential connections to Apéry's irrationality proofs for $\zeta(3)$ , and whether these specific approximations yield new irrationality or transcendence results.