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On $v_2(q),v_3(q),v_4(q)$ , and Andrews' Conjectures 5 and 6

This paper proves that the coefficients of Ramanujan's $q$ -series $v_2(q), v_3(q)$ , and $v_4(q)$ exhibit an exceptional sign pair phenomenon where infinitely many consecutive coefficients share the same sign, with at least one being a local minimum of the absolute values, by combining precise asymptotic expansions with an analysis of governing oscillatory factors.

Original authors: Jayashree Kalita

Published 2026-07-24

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Jayashree Kalita

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 world where numbers aren't just cold, static digits, but dancers in a grand, chaotic ballroom. This is the realm of number theory, a branch of mathematics that studies the hidden patterns and relationships between integers. In this specific corner of the dance floor, mathematicians are fascinated by something called $q$ -series. Think of these as special recipes or musical scores that generate an endless list of numbers (coefficients) when you expand them. For a long time, these numbers seemed to follow a strict, predictable rhythm: positive, negative, positive, negative, like a heartbeat. But in the early 20th century, the legendary mathematician Srinivasa Ramanujan left behind a "Lost Notebook" filled with these mysterious scores. When modern mathematicians looked closer, they noticed something strange: while the numbers mostly kept their alternating rhythm, occasionally, two neighbors would break the rules and dance in the same direction. The big question was: Is this just a rare glitch, or does it happen often enough to be a fundamental part of the dance?

This paper, written by Jayashree Kalita, dives deep into three specific "scores" from Ramanujan's notebook, known as $v_2(q)$ , $v_3(q)$ , and $v_4(q)$ . Building on recent work that proved these numbers mostly alternate signs (positive, negative, positive, negative), Kalita investigates the "exceptional" moments where this pattern breaks. She proves that these rule-breaking moments aren't just random accidents; they happen infinitely often and follow a very specific, structured pattern. The paper shows that whenever two numbers in a row share the same sign, at least one of them is a "local minimum"—a dip in the height of the numbers, like a valley in a mountain range. The author uses powerful mathematical tools called asymptotic expansions (which are like super-accurate maps of how the numbers behave when they get huge) to track down these valleys and prove that the "same-sign" pairs are a guaranteed feature of these sequences, not a fluke.

The Story of the Dancing Numbers

Let's meet our main characters: three sequences of numbers generated by Ramanujan's mysterious formulas. For decades, mathematicians have watched these numbers grow. They noticed a "great sign regularity," meaning the numbers mostly flip-flop between positive and negative like a pendulum swinging back and forth. If you list them out, it looks like this: $+$ , $-$, $+$ , $-$, $+$ , $-$... It's a very orderly party.

But then, a mathematician named George Andrews noticed something odd. He saw that sometimes, the pendulum gets stuck. Two numbers in a row would have the same sign. He called these "exceptional sign pairs." Imagine a line of people passing a ball; everyone throws it left, then right, then left, then right. But suddenly, two people in a row throw it to the right. Andrews guessed that this happens infinitely many times, but he couldn't prove it. He also guessed that when this happens, the numbers involved are usually "local minima." In our ball-passing analogy, this means that when two people throw the ball the same way, the person throwing it is standing in a dip, and the numbers around them are bigger.

The Detective Work

Kalita's paper is the detective work that finally solves the case. She doesn't just guess; she proves it. To do this, she uses a mathematical "magnifying glass" called an asymptotic formula. This formula is like a crystal ball that tells us exactly what the numbers will look like when they get incredibly large (approaching infinity).

The formula has a few parts. One part makes the numbers grow huge (like a balloon inflating). Another part makes them oscillate, or wiggle, like a wave. This wave is the key. The paper shows that the "same sign" pairs happen right when this wave is about to cross zero but hasn't quite gotten there yet. It's like watching a tide come in; the water level is dropping, but for a split second, it's still high enough to touch two rocks at once before dipping lower.

The Big Discovery

The paper proves two main things, and it does so with absolute certainty (mathematical proof, not just a guess):

Infinite Occurrence: There are infinitely many pairs of consecutive numbers in these sequences that have the same sign. The "glitch" isn't a one-time mistake; it's a permanent feature of the dance.
The Valley Rule: In every single one of these same-sign pairs, at least one of the numbers is a local minimum. This means that if you see two numbers with the same sign, you are guaranteed to find a "valley" right there. The numbers around them are bigger, and the pattern dips down before rising again.

The author uses a clever trick involving a "uniform separation theorem." Imagine the zeros of the wave (the points where the sign should change) are like marks on a ruler. The paper proves that these marks never land exactly on whole numbers. They always land in the gaps between the integers. Because of this, the wave spends a little bit of time hovering near zero, allowing two whole numbers to sit on the same side of the line before the wave finally flips.

Why It Matters

This isn't just about counting numbers. It resolves a set of conjectures (hunches) made by Andrews back in 1986. For the series $v_2$ , $v_3$ , and $v_4$ , the paper confirms that the "exceptional sign pair phenomenon" is real, infinite, and structured. It connects the chaotic behavior of these numbers to the geometry of waves and the arithmetic of square roots.

The paper doesn't just say "it happens." It says, "Here is exactly when it happens, here is how often, and here is the shape of the numbers when it does." It turns a mysterious observation from a notebook written a century ago into a proven fact of mathematics. So, the next time you see a sequence of numbers that seems to break the rules, remember: in the world of Ramanujan's lost notebook, breaking the rules is actually the most rule-abiding thing of all.

Technical Summary: On $v_2(q)$ , $v_3(q)$ , $v_4(q)$ , and Andrews' Conjectures 5 and 6

Problem Statement
This paper addresses the sign behavior of the coefficients of three specific $q$ -series found in Ramanujan's Lost Notebook: $v_2(q)$ , $v_3(q)$ , and $v_4(q)$ . While recent work by Kundu, Storzer, Wang, and the author established that the coefficients of these series follow an "almost alternating" sign pattern (meaning the signs alternate for a set of density one), exceptions exist. Specifically, there are indices where consecutive coefficients share the same sign.

Building on numerical observations by Andrews, the paper focuses on proving two specific conjectures regarding these exceptional indices for $v_2(q)$ , $v_3(q)$ , and $v_4(q)$ :

Conjecture 5: There exist infinitely many sequences of consecutive coefficients (specifically pairs) that share the same sign.
Conjecture 6: Within these sequences of same-sign coefficients, at least one coefficient corresponds to a local minimum of the sequence of absolute values.

These conjectures were previously proven for the series $v_1(q)$ by Bachraoui. The goal of this paper is to extend these proofs to the remaining three series, thereby resolving all of Andrews' conjectures and their analogs for the five series originally studied.

Methodology
The proof relies on a combination of precise asymptotic analysis and the study of oscillatory factors within the coefficient expansions.

Asymptotic Formulas: The authors utilize the asymptotic formulas for the coefficients $V_j(n)$ (where $j \in \{2, 3, 4\}$ ) established in their previous work (Theorem B [8]). These formulas take the form:
$V_j(n) \approx (-1)^n \alpha_j \frac{e^{2\sqrt{n}\text{Re}(\sqrt{W_j})}}{\sqrt{n}} \cos\left(2\sqrt{n}\text{Im}(\sqrt{W_j}) - \phi_j\right)$
The sign behavior is governed by the oscillatory cosine term.
General Framework: To handle the specific constants for $v_2, v_3, v_4$ uniformly, the authors define a general class of integer sequences $V(n)$ satisfying a similar asymptotic structure involving a parameter $\lambda$ (related to the imaginary part of $\sqrt{W_j}$ ) and a phase shift $\phi$ .
Uniform Separation Theorem (Section 2): A critical step is proving that the zeros of the oscillatory factor, $s_m$ , do not align too closely with the square roots of integers. The authors prove that there exists a subsequence of indices $m$ such that the distance between the zero $s_m$ and the nearest integer is uniformly bounded away from zero. This relies on the irrationality of specific constants derived from the Bloch–Wigner dilogarithm $D(e^{\pi i/3})$ .
Analysis Near Zeros (Section 3): The authors analyze the behavior of the oscillatory factor $F(x) = \cos(\lambda x + \phi)$ near its zeros. They identify integers $N_m$ (the floor of the squared zeros) and their neighbors ( $N_m-1, N_m+1, N_m+2$ ). Using Taylor expansions and bounds on the sine function near zero, they demonstrate that the magnitude of the oscillatory factor at $N_m$ and $N_m+1$ is strictly smaller than at their immediate neighbors ( $N_m-1$ and $N_m+2$ ).
Synthesis (Section 4): By combining the sign properties of the oscillatory term (which flips sign at the zeros) with the magnitude estimates, the authors show that for sufficiently large $m$ , the error terms in the asymptotic expansion are negligible compared to the main term. This allows them to conclude that the signs of $V(N_m)$ and $V(N_m+1)$ are identical, and that the absolute value of at least one of these terms is a local minimum.

Key Contributions and Results

Proof of Conjectures 5 and 6: The paper proves that for $j \in \{2, 3, 4\}$ , there exists an infinite sequence of indices $N_k$ such that $V_j(N_k)$ and $V_j(N_k+1)$ have the same sign. Furthermore, at least one of $|V_j(N_k)|$ or $|V_j(N_k+1)|$ is a local minimum of the sequence of absolute values.
General Theorem (Theorem 7): The authors establish a general result for any integer sequence satisfying the specified asymptotic form with irrational $\pi^2/\lambda^2$ . This theorem guarantees the existence of exceptional sign pairs and local minima under these conditions.
Resolution of Andrews' Conjectures: By applying Theorem 7 to the specific parameters of $v_2(q)$ , $v_3(q)$ , and $v_4(q)$ , the paper confirms that the "exceptional sign pair phenomenon" occurs infinitely often for these series, resolving the open cases of Andrews' conjectures.

Significance
The paper claims to resolve all of Andrews' conjectures regarding the growth and sign regularity of the five $q$ -series from Ramanujan's Lost Notebook ( $\sigma(q), v_1(q), v_2(q), v_3(q), v_4(q)$ ). While previous work had established the "almost alternating" nature of the signs, this work characterizes the structure of the exceptions. It demonstrates that these exceptions are not random but occur in a structured, infinite manner where consecutive coefficients share a sign and exhibit local minima in magnitude. The methodology provides a rigorous framework for analyzing sign patterns in $q$ -series coefficients driven by oscillatory asymptotic terms involving the Bloch–Wigner dilogarithm.

The Story of the Dancing Numbers

The Detective Work

The Big Discovery

Why It Matters

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