Keith's congruence conjecture for coefficients of a reciprocal false theta function

Let $c_5(n)$ be defined by

\frac{1}{\Psi(-q^5,q)}=\sum_{n=0}^\infty c_5(n)q^n,

where

\Psi(a,b):=\sum_{n=0}^\infty a^{n(n+1)/2}b^{n(n-1)/2}-\sum_{n=-\infty}^{-1}a^{n(n+1)/2}b^{n(n-1)/2}.

Keith's conjecture. For every integer $n\geq 0$ ,

c_5(32n+31)\equiv 0\pmod 8,\qquad c_5(128n+123)\equiv 0\pmod 8,

c_5(512n+491)\equiv 0\pmod 8,\qquad c_5(64n+19)\equiv 0\pmod 4,

c_5(256n+75)\equiv 0\pmod 4,

and, for $i\in\{110,138,194\}$ and $j\in\{19,47,75,103,159,187\}$ ,

c_5(196n+i)\equiv 0\pmod 4,\qquad c_5(196n+j)\equiv 0\pmod 4.

These congruences concern arithmetic progressions in the coefficients of the reciprocal of a false theta function. The conjecture was posed by Keith and is the result proved by the paper from which this candidate is extracted; the supplied parser status is unknown, so the database status remains open pending explicit resolution evidence.

References

Primary source

Jing Jin, Sijia Wang and Olivia X. M. Yao, “Proof of a conjecture of Keith on congruences of the reciprocal of a false theta function”, arXiv:2508.01532 (2025).

Posted 1 month ago

Progress summary

Refreshed 1 month ago

Claimed solved

A paper posted on August 3, 2025 claims to prove all of Keith’s coefficient divisibility predictions and a broader family, but the proof has not been independently verified.

Keith posed these congruences for the coefficients $c_5(n)$ of a reciprocal false theta function in work dated January 4, 2025.

Known results

Keith proved $c_5(8n+5)\equiv 0\pmod 2$ .
Keith proved $c_5(32n+31)\equiv 0\pmod 4$ .

August 3, 2025 claimed proof

Jing Jin, Sijia Wang, and Olivia X.M. Yao’s arXiv preprint Proof of a conjecture of Keith on congruences of the reciprocal of a false theta function asserts all listed congruences, including the remaining progressions $196n+j$ , and proves a generalization for primes $p\equiv 7\pmod 8$ . No retrieved source reports a counterexample, withdrawal, or verification.

Current status (as of September 2026): Keith’s conjecture is claimed proved by the August 2025 preprint, but remains unverified in the retrieved record.

Sources

No solutions have been posted yet.