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Sign law for Ramanujan's third order mock theta function $\rho(q)$

This paper proves that the coefficients $r(n)$ of Ramanujan's third-order mock theta function $\rho(q)$ exhibit a specific sign pattern for sufficiently large $n$ , where $r(3n)$ is positive and $r(3n+1), r(3n+2)$ are negative, by deriving an asymptotic formula using Watson's relation and Rademacher-type expansions.

Original authors: Manosij Ghosh Dastidar

Published 2026-06-29

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CC BY 4.0

Original authors: Manosij Ghosh Dastidar

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, rhythmic drumbeat. This drumbeat is made of numbers, and mathematicians have been trying to figure out the pattern of its "loudness" (whether the numbers are positive or negative) for a long time.

This paper, written by Manosij Ghosh Dastidar, is about solving a specific mystery regarding a famous musical pattern discovered by the legendary Indian mathematician Srinivasa Ramanujan. This pattern is called a "mock theta function," specifically one named $\rho(q)$ (rho).

Here is the story of what the paper does, explained simply:

1. The Mystery: A Rigid Rule

When mathematicians list out the numbers generated by Ramanujan's pattern, they noticed something strange. It looks like the numbers follow a strict "traffic light" rule based on their position:

Green Light (Positive): If the number is in a position divisible by 3 (like 3rd, 6th, 9th), it is usually positive (a loud, forward beat).
Red Light (Negative): If the number is in a position that is 1 or 2 steps past a multiple of 3, it is usually negative (a backward or quiet beat).

There were a few early glitches where the drumbeat went silent (the number was zero), but the pattern seemed incredibly strong. The author wanted to prove that this rule isn't just a coincidence for small numbers, but a law that holds true forever as the numbers get huge.

2. The Detective Work: Breaking the Code

To prove this, the author didn't just look at the numbers one by one. Instead, he used a "magic key" discovered by another mathematician named Watson.

Think of the pattern $\rho(q)$ as a locked box. Watson found a way to open it by relating it to two other things:

Another similar pattern called $\omega(q)$ .
A "Theta-Eta product" (let's call it $T(q)$ ), which is a well-understood mathematical object, like a standard clock.

The relationship is like a balance scale:
$2 \times (\text{Our Mystery Pattern}) + (\text{Another Pattern}) = (\text{The Standard Clock})$

By rearranging this, the author could calculate the mystery numbers by subtracting the "Another Pattern" from the "Standard Clock."

3. The Telescope: Seeing the Distant Future

The author used a powerful mathematical tool called a Rademacher expansion. Imagine this as a high-powered telescope.

When you look at the numbers with the naked eye, they look chaotic.
When you look through the telescope, you see that the numbers are actually made of two main parts:
1. A Huge Wave (the main term) that grows very fast.
2. A Tiny Ripple (the error term) that is so small it barely matters when the numbers get big.

The author calculated the "Huge Wave" for both the "Standard Clock" and the "Another Pattern."

4. The Big Reveal: The Cancellation

Here is the clever part of the proof:

The "Huge Wave" from the Standard Clock and the "Huge Wave" from the Other Pattern were almost identical.
When the author subtracted them (as the balance scale required), these two massive waves cancelled each other out perfectly.
This left behind only the next biggest wave (the second layer of the telescope).

It turns out that this remaining wave has a specific "color" (sign) depending on where you are in the sequence:

If you are at a multiple of 3, the wave is Positive.
If you are at the other positions, the wave is Negative.

Because this remaining wave is much larger than the tiny ripples (the error), the sign of the final number is determined entirely by this wave.

5. The Conclusion

The paper proves that for all sufficiently large numbers, the "traffic light" rule is absolutely true:

Positions divisible by 3: Always Positive.
Other positions: Always Negative.

The author also checked the early numbers (the "glitches") and found that the rule holds true for almost all of them, except for a tiny handful of zeros (like the 2nd, 4th, 8th, 11th, and 20th numbers). The paper conjectures that after the 20th number, the rule is perfect with no exceptions.

In short: The author used a mathematical "balance scale" and a "telescope" to show that Ramanujan's mysterious drumbeat follows a strict, predictable rhythm of positive and negative numbers that never breaks once the music gets loud enough.

Technical Summary: Sign Law for Ramanujan's Third Order Mock Theta Function $\rho(q)$

Problem Statement
This paper investigates the coefficients $r(n)$ of Ramanujan's third-order mock theta function $\rho(q)$ , defined by the series:
$\rho(q) = \sum_{m \ge 0} \frac{q^{2m(m+1)}}{(1 + q + q^2)(1 + q^3 + q^6) \cdots (1 + q^{2m+1} + q^{4m+2})} = \sum_{n \ge 0} r(n)q^n.$
Numerical computation of the initial coefficients reveals a rigid sign pattern dependent on $n \pmod 3$ :

$r(3n) > 0$
$r(3n+1) \le 0$
$r(3n+2) \le 0$

While early terms in the negative residue classes occasionally vanish (specifically $r(2), r(4), r(8), r(11), r(20) = 0$ ), no terms with the "wrong" sign have been observed. The paper aims to prove the asymptotic validity of this sign law.

Methodology
The proof relies on the circle method and asymptotic analysis of mock theta coefficients, specifically utilizing the following steps:

Watson's Identity: The authors employ a classical identity by Watson relating $\rho(q)$ to another mock theta function $\omega(q)$ and a theta-eta product $T(q)$ :
$2\rho(q) + \omega(q) = T(q).$
In terms of coefficients, this yields the relation $2r(n) = t(n) - w(n)$ , where $t(n)$ and $w(n)$ are the coefficients of $T(q)$ and $\omega(q)$ , respectively.
Rademacher-Type Expansions:
- For $\omega(q)$ : The authors utilize existing Rademacher-type expansions for $\omega(q)$ (based on work by Garthwaite and Andersen–Anderson). They explicitly calculate the first three Kloosterman sums ( $c=1, 2, 3$ ) to determine the leading asymptotic terms.
- For $T(q)$ : The function $T(q)$ is expressed as a theta-eta quotient: $T(q) = 3 \psi(q^3)^2 / (q^2; q^2)_\infty$ . The authors derive a new Rademacher-type expansion for this eta-quotient. This involves computing the local asymptotic behavior of $T(q)$ near roots of unity ( $q=1, -1, \zeta, \zeta^2$ ) to determine the corresponding multiplier system and Bessel coefficients.
Cancellation and Comparison:
By substituting the expansions into $2r(n) = t(n) - w(n)$ , the authors observe that the leading terms (corresponding to $c=1$ ) cancel exactly. The $c=2$ terms vanish for both functions. Consequently, the asymptotic behavior of $r(n)$ is governed by the $c=3$ terms.
Sign Determination:
The authors calculate the specific constants arising from the $c=3$ terms for both $t(n)$ and $w(n)$ . They demonstrate that the resulting coefficient for $r(n)$ is proportional to a constant $\kappa_{n \pmod 3}$ multiplied by a modified Bessel function $I_{1/2}$ . Since $I_{1/2}(x) > 0$ for $x > 0$ , the sign of $r(n)$ for large $n$ is determined entirely by the sign of $\kappa_{n \pmod 3}$ .

Key Results
The paper establishes the following asymptotic formula for $r(n)$ as $n \to \infty$ :
$r(n) \sim \kappa_{n \pmod 3} \frac{2\pi}{(12n+8)^{1/4}} I_{1/2}\left( \frac{\pi\sqrt{12n+8}}{18} \right),$
where the constants are:

$\kappa_0 = \frac{1}{3} \cos(\frac{\pi}{18}) > 0$
$\kappa_1 = -\frac{1}{3} \sin(\frac{2\pi}{9}) < 0$
$\kappa_2 = -\frac{1}{3} \sin(\frac{\pi}{9}) < 0$

Theorem 1.2 states that for all sufficiently large $n$ :

$r(3n) > 0$
$r(3n+1) < 0$
$r(3n+2) < 0$

The error term in this approximation is shown to be exponentially smaller than the main term ( $O(e^{\pi\sqrt{12n+8}/24})$ vs. main term growth $e^{\pi\sqrt{12n+8}/18}$ ), ensuring the sign is eventually fixed.

Significance and Claims
The paper claims to provide the first proof of the asymptotic sign law for the coefficients of $\rho(q)$ . The work fits within the broader context of understanding the arithmetic properties of mock theta functions, extending the circle of ideas from Dragonette, Andrews, Bringmann, and Ono.

The authors explicitly state that their proof covers the "eventual" behavior of the coefficients. They do not claim to determine the exact finite cutoff for the sign law to hold for all $n$ , though experimental data suggests the strict inequality holds for all $n \ge 21$ (with the noted zeros in the negative classes). The paper concludes by conjecturing that similar sign laws may exist for other third-order mock theta functions, specifically $\phi(q)$ and $\chi(q)$ , though these are not proven in this work.