Parity results for the reciprocals of false theta functions

TL;DR

The paper investigates parity (mod 2) properties of the coefficients c_{r,s}(n) defined by reciprocals of false theta functions, leveraging Ramanujan's general theta function identities. By transforming generating functions modulo 2 via f(a,b) relations and extracting subsequences, it derives infinite families of parity vanishing for c_{9,1}, c_{7,3}, c_{13,1}, c_{11,3}, and c_{9,5}, including several results that confirm Keith's conjectures. The approach yields structured 2-adic progressions tied to powers of 2 and to primes in residue classes (notably p ≡ 11,19 mod 20 and p ≡ 15,27 mod 28), linking parity to representations by quadratic forms and to modular-type identities. These findings enrich the arithmetic understanding of false theta reciprocals and provide broad parity criteria that align with and extend Keith's conjectures.

Abstract

Recently, Keith investigated arithmetic properties for the reciprocals of some false theta functions and posed several conjectures. In this paper, we prove some parity results for the reciprocals of some false theta functions by using some identities on Ramanujan's general theta function. In particular, our results imply some conjectures of Keith.

Theorems & Definitions (14)

• Conjecture 1.1
• Theorem 2.1
• Theorem 3.1
• Theorem 4.1
• Corollary 4.2
• Theorem 4.3
• Theorem 4.4
• Corollary 4.5
• Theorem 5.1
• Theorem 5.2