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Proofs for Andrews' Conjectures 5 and 6 on $v_1(q)$

This paper provides unconditional proofs for Andrews' Conjectures 5 and 6 regarding the coefficients of $v_1(q)$ by analyzing the simple zeros of a trigonometric factor in the asymptotic expansion and demonstrating that a specific quadratic sequence remains a positive distance from the integers infinitely often.

Original authors: Mohamed El Bachraoui

Published 2026-04-10

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Mohamed El Bachraoui

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 listening to a very complex, chaotic piece of music. The notes are generated by a mysterious mathematical formula called a q-series. If you write down the volume of each note (the coefficients), you get a long list of numbers: some are loud and positive, some are soft and negative, and they seem to jump around wildly.

This is the story of a specific musical sequence called $v_1(q)$ , studied by the famous mathematician George Andrews. He noticed something strange: while the volume of the notes seemed chaotic, there might be hidden patterns in how the signs (positive or negative) behave.

This paper by Mohamed El Bachraoui is like a detective story. It solves two specific mysteries (Conjectures 5 and 6) about how these numbers behave when you get very far out in the sequence.

Here is the breakdown of the paper using simple analogies:

1. The Mystery: The "Sign" of the Notes

Andrews made two guesses about the long-term behavior of these numbers:

Conjecture 5: He guessed that if you look far enough out, you will eventually find a "trio" of notes that all have the same sign (three positives in a row, or three negatives in a row). Before this, the signs were thought to flip-flop too quickly to allow this.
Conjecture 6: He also guessed that in these trios, the middle note (or one of the ends) would be a "local minimum." Imagine a valley in a mountain range: the notes on either side are "higher" (louder), and this specific note is a quiet dip.

2. The Map: The Asymptotic Formula

To solve this, the author uses a "map" created by other mathematicians (Folsom, Males, Rolen, and Storzer). This map is an asymptotic formula.

The Analogy: Imagine trying to predict the weather. You can't predict every single raindrop, but you can predict the general trend: "It will be hot and humid."
The formula tells us that the numbers $V_1(n)$ behave like a giant wave that is growing exponentially (getting huge) but is also oscillating (waving up and down). The wave is controlled by a "trigonometric factor" (like a sine wave) that decides whether the number is positive or negative.

3. The Problem: The Wave and the Integers

The tricky part is that the "wave" moves at a specific speed determined by a number called $\alpha$ (which is related to a famous constant called the Bloch-Wigner dilogarithm).

The author needs to find specific points on the number line (integers) where this wave is in a "sweet spot."
Specifically, he needs to find integers where the wave is close to a "zero" (where it flips from positive to negative) but not exactly on the zero.
The Metaphor: Imagine a tightrope walker (the wave) crossing a canyon. The "zeros" are the points where the walker is perfectly balanced on a pole. The author needs to find spots where the walker is almost on the pole, but slightly to the left or right, so that three steps in a row land on the same side of the pole.

4. The Solution: Proving the "Sweet Spots" Exist

The core of the paper is proving that these "sweet spots" actually exist infinitely often.

Step 1: The Distance Check (Theorem 3)
The author proves that the wave's rhythm ( $\alpha$ ) is "irrational" (it doesn't repeat in a simple pattern). Because of this, the wave will eventually land at a safe distance from the "poles" (integers) infinitely many times. It won't get stuck exactly on the edge; it will always stay a little bit away. This guarantees that we can find our "trios."
Step 2: The Trio Formation (Theorems 4 & 5)
Once we know the wave stays away from the poles, the author shows that if you pick a spot where the wave is just about to cross zero, the three numbers immediately before and after it will all be on the same side of the line.
- The Result: You get a sequence like $+100, +105, +110$ (all positive) or $-50, -55, -60$ (all negative).
Step 3: The Valley (Local Minima)
The author also proves that at the edges of these trios, the numbers dip down.
- The Analogy: Think of a "W" shape. The middle of the "W" is a peak, but the two outer dips are "local minima." The paper proves that the numbers at the start and end of these trios are quieter (smaller in absolute value) than their immediate neighbors.

5. The Final Verdict

The paper concludes by combining the math with a computer check of the first few numbers.

The "Real World" Check: The author calculated the first few terms manually (like checking the first few pages of a book) and found that the numbers 293, 410, 545, and 702 already follow the rules.
The Infinite Future: Using the math from the first half of the paper, he proves that this pattern doesn't just happen at the start; it happens forever for larger and larger numbers.

Summary

In plain English, this paper says:

"We looked at a chaotic sequence of numbers. We proved that, despite the chaos, there are infinitely many places where three numbers in a row have the same sign (all positive or all negative). Furthermore, at the edges of these groups, the numbers dip down to form a 'valley' compared to their neighbors. We proved this happens forever, not just by guessing, but by showing that the mathematical 'wave' governing these numbers inevitably creates these patterns."

It's a victory for order over chaos, showing that even in the most complex mathematical sounds, there is a hidden, rhythmic structure.

1. Problem Statement

The paper addresses two specific conjectures proposed by George Andrews regarding the coefficients $V_1(n)$ of the $q$ -series $v_1(q)$ , which originates from Ramanujan's lost notebook:
$v_1(q) = \sum_{n \ge 0} \frac{q^{n(n+1)/2}}{(-q^2; q^2)_n} = \sum_{n \ge 0} V_1(n)q^n$

While recent work by Folsom, Males, Rolen, and Storzer had proven Conjecture 4 (regarding the sign distribution of four consecutive coefficients) and a density-one version of Conjecture 3 (regarding the growth of $|V_1(n)|$ ), Conjectures 5 and 6 remained open.

Conjecture 5: There exists an infinite sequence of integers $N_n$ (starting with 293, 410, 545, 702, ...) such that for $n \ge 5$ , the three consecutive coefficients $V_1(N_n), V_1(N_n+1), V_1(N_n+2)$ share the same sign.
Conjecture 6: For the same sequence, the absolute values $|V_1(N_n)|, |V_1(N_n+1)|, |V_1(N_n+2)|$ contain a local minimum. Specifically, the paper aims to show that the endpoints of these triples are strict local minima of the sequence $|V_1(j)|$ .

2. Methodology

The proof relies on a rigorous asymptotic analysis of the coefficients $V_1(n)$ combined with number-theoretic properties of quadratic sequences.

A. Asymptotic Expansion

The author utilizes the asymptotic formula established by Folsom et al. (Theorem 2), which approximates $V_1(n)$ as $n \to \infty$ :
$V_1(n) \approx (-1)^{\lfloor n/2 \rfloor} \frac{e^{c\sqrt{n}}}{\sqrt{n}} \left( \gamma_+ + (-1)^n \gamma_- \right) \left( \cos(c\sqrt{n}) - (-1)^n \sin(c\sqrt{n}) \right)$
where $c = \sqrt{2|V|}$ and $V$ is related to the Bloch–Wigner dilogarithm.
The author separates this into even and odd cases, defining trigonometric factors $F_-(x) = \cos x - \sin x$ and $F_+(x) = \cos x + \sin x$ . The sign and magnitude of $V_1(n)$ are dominated by these trigonometric terms evaluated at $c\sqrt{n}$ .

B. Analysis of Zeros and Quadratic Sequences

The proof identifies the zeros of the trigonometric factors:

Zeros of $F_-$ occur at $r_m = \pi(m + 1/4)$ .
Zeros of $F_+$ occur at $s_m = \pi(m + 3/4)$ .

The author defines a constant $\alpha = \pi^2 / c^2$ . The problem reduces to analyzing the quadratic sequences:
$P_-(m) = \alpha \left(m + \frac{1}{4}\right)^2 \quad \text{and} \quad P_+(m) = \alpha \left(m + \frac{3}{4}\right)^2$
The core technical challenge is to show that these sequences stay a positive distance away from the integers infinitely often. This ensures that the argument $c\sqrt{n}$ stays close to the zeros of the trigonometric functions without hitting them exactly (which would make the coefficient zero) or landing in a region where the sign flips unpredictably.

C. Number-Theoretic Argument (Theorem 3)

The author proves that there exist infinite sets of integers $M_-$ and $M_+$ such that the fractional parts of $P_-(m)$ and $P_+(m)$ are bounded away from 0 and 1.

Case 1 ( $\alpha$ irrational): By Weyl's equidistribution theorem, the sequences are equidistributed modulo 1, guaranteeing infinitely many points far from integers.
Case 2 ( $\alpha$ rational): The author proves $\alpha$ is not an integer (Lemma 1) and shows the sequences are periodic modulo 1. Since not all terms are integers, the minimum distance to an integer over the period is strictly positive.

D. Construction of the Sequence

Using the sets $M_-$ and $M_+$ , the author constructs specific integers $N_m$ (for even cases) and $M_m$ (for odd cases) that bracket the values $\alpha(m+1/4)^2$ and $\alpha(m+3/4)^2$ .

For $m \in M_-$ , $N_m$ is the unique even integer such that $N_m < \alpha(m+1/4)^2 < N_m+2$ .
The author then analyzes the signs of $V_1(N_m), V_1(N_m+1), V_1(N_m+2)$ using the asymptotic expansion and error bounds, proving they share the same sign.
Simultaneously, the magnitude analysis shows that $|V_1(N_m)|$ and $|V_1(N_m+2)|$ are significantly smaller than their neighbors, establishing them as strict local minima.

3. Key Contributions

Unconditional Proof: The paper provides a rigorous, unconditional proof of Conjectures 5 and 6, removing the need for heuristic arguments previously used by other researchers.
Explicit Construction: It explicitly constructs the infinite sequence $N_n$ satisfying the conjectures, starting with the specific values $293, 410, 545, 702$ and extending to $N_n > 10n^2$ for $n \ge 9$ .
Refined Asymptotic Analysis: The paper details how the interplay between the exponential growth $e^{c\sqrt{n}}$ , the polynomial decay $1/\sqrt{n}$ , and the oscillating trigonometric factors creates "windows" of same-sign coefficients and local minima.
Number-Theoretic Lemma: It establishes a specific lemma regarding the distance of quadratic sequences $\alpha(m+k)^2$ from the integers, which is crucial for controlling the sign of the coefficients.

4. Results

Theorem 1: Confirms the existence of a sequence $(N_n)_{n \ge 5}$ $(N_{n})_{n \geq 5}$ such that:
- $N_5=293, N_6=410, N_7=545, N_8=702$ .
- $N_n > 10n^2$ for $n \ge 9$ .
- $V_1(N_n), V_1(N_n+1), V_1(N_n+2)$ have the same sign.
- The set $\{|V_1(N_n)|, |V_1(N_n+1)|, |V_1(N_n+2)|\}$ contains a local minimum (specifically, the endpoints $N_n$ and $N_n+2$ are strict local minima of the sequence $|V_1(j)|$ ).
Verification of Initial Values: The paper computationally verifies the conjectures for the first four terms (293, 410, 545, 702) by direct expansion of the series.

5. Significance

This work completes the resolution of the major conjectures regarding the sign regularity and magnitude behavior of the coefficients of $v_1(q)$ .

Resolution of Open Problems: It closes the gap left by the 2024/2025 work of Folsom et al., providing a full proof for the existence of infinite families of same-sign triples and local minima.
Methodological Insight: The paper demonstrates how deep properties of the Bloch–Wigner dilogarithm (via the constant $\alpha$ ) and Diophantine approximation (distance to integers) dictate the fine-scale structure of $q$ -series coefficients.
Ramanujan's Legacy: By rigorously proving Andrews' conjectures derived from Ramanujan's lost notebook, the paper reinforces the deep connections between mock theta functions, partition theory, and analytic number theory.

In summary, El Bachraoui successfully bridges asymptotic analysis and number theory to prove that the coefficients of $v_1(q)$ exhibit a highly structured pattern of sign regularity and local minima, confirming Andrews' intuition with mathematical precision.