Sign Laws and Mock Theta Functions

Manosij Ghosh Dastidar

arxiv: 2606.31606 · v1 · pith:TBJTKQQGnew · submitted 2026-06-30 · 🧮 math.NT

Sign Laws and Mock Theta Functions

Manosij Ghosh Dastidar This is my paper

Pith reviewed 2026-07-01 04:02 UTC · model grok-4.3

classification 🧮 math.NT

keywords mock theta functionssign patternsRamanujanroot-of-unity estimateseta quotientsWatson's identityq-series coefficients

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The pith

Ramanujan's third-order mock theta function has coefficients whose signs are completely determined by residue class modulo 3, except for five small exceptions.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper proves that the coefficients r(n) of Ramanujan's mock theta function ρ(q) satisfy r(3m) > 0 while r(3m+1) ≤ 0 and r(3m+2) ≤ 0, with equality only at n = 2, 4, 8, 11, 20. Watson's identity reduces the problem to bounding the difference between another mock theta function ω(q) and an explicit eta quotient T(q). Effective estimates at roots of unity show that polar terms cancel at q = 1, the term at q = -1 remains polynomially bounded, and the leading contribution at primitive cubic roots of unity carries the observed sign pattern. Direct verification handles the finite range, completing the proof for all n.

Core claim

The sign law r(3m) > 0, r(3m+1) ≤ 0, r(3m+2) ≤ 0 holds with equality precisely at n=2,4,8,11,20. This is obtained by showing that the difference ω(q) − T(q) has polar contributions that cancel at q=1, remains polynomially bounded at q=−1, and receives its first surviving exponential term at the primitive cubic roots of unity, where the coefficients are κ0 = (1/3) cos(π/18) > 0, κ1 = −(1/3) sin(2π/9) < 0, and κ2 = −(1/3) sin(π/9) < 0; the resulting asymptotic together with integer verification establishes the pattern everywhere.

What carries the argument

Watson's identity 2ρ(q) + ω(q) = T(q) combined with root-of-unity estimates isolating the dominant cubic-root contribution after cancellation at q=1.

If this is right

The same root-of-unity method yields sign laws for the other third-order mock theta functions φ(q) and χ(q).
The asymptotic sign is governed solely by the three explicit constants κ0, κ1, κ2 once n exceeds a finite bound.
Exact integer-arithmetic checks suffice to settle all remaining small cases after the asymptotic takes over.
Cancellation of polar terms at q=1 and polynomial control at q=−1 are the precise conditions that isolate the cubic-root dominance.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The technique may produce sign laws for coefficients of other q-series whose modular transformations involve similar eta quotients.
The pattern suggests that generating functions built from these mock theta functions inherit arithmetic positivity properties modulo 3.
Numerical verification of the predicted signs for n up to several hundred thousand would provide an independent consistency check on the effective constants.
Analogous cancellation arguments could be tested on higher-order mock theta functions where the relevant roots of unity are of higher order.

Load-bearing premise

The polar parts at q=1 cancel exactly while the contribution at q=−1 grows at most polynomially, allowing the cubic-root term to fix the sign for all sufficiently large n.

What would settle it

An explicit computation of r(n) for some n > 20 that violates r(3m) > 0 or the two non-positive conditions, or an analytic demonstration that a different root-of-unity term overtakes the cubic contribution.

read the original abstract

Let \[ \rho(q)=\sum_{m\geq 0}\frac{q^{2m(m+1)}}{(1+q+q^2)(1+q^3+q^6)\cdots(1+q^{2m+1}+q^{4m+2})} =\sum_{n\geq 0}r(n)q^n \] be Ramanujan's third order mock theta function. We prove the sign law \[ r(3m)>0,\qquad r(3m+1)\leq 0,\qquad r(3m+2)\leq 0, \] with equality precisely at $n=2,4,8,11,20$. Watson's identity \[ 2\rho(q)+\omega(q)=T(q) \] reduces the problem to comparing the mock theta function $\omega(q)$ with the eta quotient \[ T(q)=3\frac{(q^6;q^6)_\infty^4}{(q^3;q^3)_\infty^2(q^2;q^2)_\infty}. \] We prove effective root-of-unity estimates for this difference. The polar contributions at $q=1$ cancel, the contribution at $q=-1$ is polynomially bounded, and the first surviving exponential term occurs at the primitive cubic roots of unity. It has the sign pattern \[ \kappa_0=\frac13\cos\frac\pi{18}>0,\qquad \kappa_1=-\frac13\sin\frac{2\pi}{9}<0, \qquad \kappa_2=-\frac13\sin\frac\pi9<0. \] The resulting effective asymptotic proves the desired sign law for all sufficiently large $n$, and an exact integer-arithmetic verification completes the finite range. We conclude by indicating how the same root-of-unity method should lead to analogous sign laws for other third order mock theta functions, including $\phi(q)$ and $\chi(q)$.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper proves the sign law for r(n) in Ramanujan's ρ(q) via Watson's identity and effective root-of-unity estimates on the difference with an eta quotient.

read the letter

The main result is a proof that r(3m) > 0 while r(3m+1) and r(3m+2) are ≤ 0, with equality only at n=2,4,8,11,20. Watson's identity reduces the problem to comparing the mock theta ω(q) against the eta quotient T(q), after which root-of-unity analysis produces an effective asymptotic whose leading term at the primitive cubic roots determines the sign pattern for large n. The polar terms cancel at q=1, the q=-1 contribution stays polynomially bounded, and the explicit constants κ0 > 0, κ1 < 0, κ2 < 0 fix the signs. Exact integer arithmetic then settles the finite range.

This is new: the specific sign law for this function and the detailed effective estimates using the cubic roots are not in the earlier literature on mock theta functions. The argument stays within the series definitions and identities, with no fitted parameters or circular steps. The outline is self-contained and the finite verification is described as exact arithmetic, which is a solid way to close the proof.

The soft spots are modest. The paper only indicates how the same method might apply to φ(q) and χ(q); those cases are not carried out. The effectiveness claim (how large n must be before the asymptotic dominates) rests on the stated bounds being sufficient, which would need verification in the full text but is asserted to work. No internal contradiction or uncontrolled remainder appears in the strategy.

The work is for specialists in q-series and mock modular forms who care about coefficient signs. A reader already familiar with Watson-type identities and root-of-unity filtrations will get the most from it. The central claim is grounded enough to deserve referee time.

Referee Report

0 major / 1 minor

Summary. The paper proves the sign law r(3m)>0, r(3m+1)≤0, r(3m+2)≤0 for the coefficients of Ramanujan's third-order mock theta function ρ(q), with equality only at n=2,4,8,11,20. Using Watson's identity 2ρ(q)+ω(q)=T(q), the proof reduces the problem to effective root-of-unity estimates on the difference ω(q)−T(q). It shows cancellation of polar terms at q=1, polynomial boundedness at q=-1, and that the leading exponential term at primitive cubic roots of unity has signs κ0>0, κ1<0, κ2<0, yielding an asymptotic that dominates for large n; the finite range is settled by exact integer-arithmetic verification. The paper indicates the method extends to other third-order mock theta functions.

Significance. If the effective estimates are valid, the result establishes a precise sign pattern for the mock theta coefficients via analytic methods with explicit constants and finite verification, rather than conjecture or numerical evidence alone. The root-of-unity approach with documented cancellations and leading-term signs provides a template for sign laws in other q-series and mock modular forms, as suggested for φ(q) and χ(q).

minor comments (1)

The abstract and introduction would benefit from a brief statement of the range up to which the finite verification was performed (e.g., the explicit N0 beyond which the asymptotic dominates).

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary of our work and for recommending acceptance. We are pleased that the analytic approach via Watson's identity and effective root-of-unity estimates is viewed as providing a useful template for sign laws in other mock theta functions.

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The paper's derivation begins from the explicit q-series definition of ρ(q), invokes the known Watson identity 2ρ(q) + ω(q) = T(q) to reduce to an eta-quotient difference, then derives effective asymptotics via root-of-unity analysis showing explicit cancellation of polar terms at q=1, polynomial boundedness at q=-1, and leading exponential terms at primitive cubic roots with explicitly computed signs κ0>0, κ1<0, κ2<0. The resulting asymptotic dominates for large n; the finite range is settled by direct integer verification. No parameter is fitted and then relabeled as a prediction, no self-citation supplies a load-bearing uniqueness theorem, and no ansatz or renaming is smuggled in. The chain is therefore self-contained against the series definitions and standard q-series identities.

Axiom & Free-Parameter Ledger

0 free parameters · 3 axioms · 0 invented entities

The proof relies on established analytic techniques for q-series and modular forms without introducing new free parameters or entities.

axioms (3)

standard math Properties of the Dedekind eta function and its quotients.
Used to define T(q).
domain assumption Watson's identity 2ρ(q) + ω(q) = T(q).
Reduces the sign problem to the difference with T(q).
standard math Analytic properties allowing root-of-unity estimates for q-series.
Basis for the effective asymptotic.

pith-pipeline@v0.9.1-grok · 5878 in / 1491 out tokens · 96533 ms · 2026-07-01T04:02:41.552704+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

9 extracted references · 1 canonical work pages

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G. E. Andrews, A. Dixit, and A. J. Yee,Partitions associated with the Ramanujan/Watson mock theta functionsω(q),ν(q)andϕ(q), Research in the Mathematical Sciences2(2015), Article 19

2015

[2] [2]

Bringmann and K

K. Bringmann and K. Ono,Thef(q)mock theta function conjecture and partition ranks, Inventiones Mathematicae165(2006), 243–266

2006

[3] [3]

Bringmann and K

K. Bringmann and K. Ono,Coefficients of harmonic Maass forms, In:Partitions,q-series, and modular forms, Developments in Mathematics23, Springer, 2012, 23–38. 28

2012

[4] [4]

Folsom,Asymptotics and Ramanujan’s mock theta functions, Philosophical Transactions of the Royal Society A378(2020), 20180448

A. Folsom,Asymptotics and Ramanujan’s mock theta functions, Philosophical Transactions of the Royal Society A378(2020), 20180448

2020

[5] [5]

Garvan and A

F. Garvan and A. Mukhopadhyay,Transformation of third order mock theta functions and newq-series identities, arXiv:2510.21157

work page arXiv

[6] [6]

G. N. Watson,The final problem: an account of the mock theta functions, Journal of the London Mathematical Society11(1936), 55–80

1936

[7] [7]

Zagier,Ramanujan’s mock theta functions and their applications (after Zwegers and Ono- Bringmann), S´ eminaire Bourbaki, Ast´ erisque326(2009), 143–164

D. Zagier,Ramanujan’s mock theta functions and their applications (after Zwegers and Ono- Bringmann), S´ eminaire Bourbaki, Ast´ erisque326(2009), 143–164

2009

[8] [8]

S. P. Zwegers,Mock theta functions, Ph.D. thesis, Universiteit Utrecht, 2002. G. N. Watson,The Final Problem: An Account of the Mock Theta Functions, Journal of the London Mathematical Society11(1936), 55–80

2002

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S. P. Zwegers,Mockϑ-functions and real analytic modular forms, In:q-Series with Appli- cations to Combinatorics, Number Theory, and Physics, Contemporary Mathematics291, American Mathematical Society, 2001, 269–277. 29

2001