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The Algebra of Parity-Twisted Crank Moments and a Prime-Detecting Expression
Abstract
We determine the algebra generated by the normalized parity-twisted crank moments and describe it explicitly as a proper subalgebra of the quasi-modular forms on Γ0(2) closed under D=qd/dq. We derive congruences from differential identities and construct a prime-detecting expression involving only the normalized twisted second moment, its products, and its derivatives. Its coefficient of qn vanishes if and only if n is prime, for every n≥2. We prove that its highest weight, eight, is minimal among prime-detecting elements of this algebra. Finally, we identify the same algebra as the one generated by MacMahon's functions Bk, obtaining a corresponding prime-detecting expression in B1.
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BibTeX
@misc{kang2026algebra,
title = {{The Algebra of Parity-Twisted Crank Moments and a Prime-Detecting Expression}},
author = {Soon-Yi Kang},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.37301},
eprint = {2609.37301},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.37301},
}
RIS
TY - UNPB
TI - The Algebra of Parity-Twisted Crank Moments and a Prime-Detecting Expression
AU - Soon-Yi Kang
PY - 2026
DA - 2026/09/29
PB - arXiv
DO - 10.48550/arxiv.2609.37301
UR - https://arxiv.org/abs/2609.37301
AB - We determine the algebra generated by the normalized parity-twisted crank moments and describe it explicitly as a proper subalgebra of the quasi-modular forms on Γ_0(2) closed under D=qd/dq. We derive congruences from differential identities and construct a prime-detecting expression involving only the normalized twisted second moment, its products, and its derivatives. Its coefficient of q^n vanishes if and only if n is prime, for every n≥2. We prove that its highest weight, eight, is minimal among prime-detecting elements of this algebra. Finally, we identify the same algebra as the one generated by MacMahon's functions B_k, obtaining a corresponding prime-detecting expression in B_1.
ER -