Prime detecting quasi-modular forms in higher level
Ben Kane, Krishnarjun Krishnamoorthy, Yuk-Kam Lau
TL;DR
This work extends the prime-detection paradigm for quasi-modular forms from full level to higher levels, introducing and exploiting the sieving operator $S_{M,m}$ to isolate arithmetic-progressions. It shows that, after removing old-forms via sieving, the surviving components are governed by Eisenstein data, and provides an explicit spanning set built from derivatives of Eisenstein series $E_{k,\chi,\psi}$ using the $H_{k,\ell,\chi,\psi}$ construction. The authors prove a progression-wise vanishing/sign-stability phenomenon and establish a finite-check criterion for prime-detection in arithmetic progressions, together with an analytic level-one appendix connecting divisor sums and zeta-quotients to prime-detecting identities. These results deepen the structural understanding of prime-detecting quasi-modular forms and enable practical verification within arithmetic progressions, with implications for partition-theoretic prime identities.
Abstract
In a previous work, the authors resolved a conjecture about the structure of prime-detecting quasi-modular forms by studying sign changes occurring in quasi-modular cusp forms. In this paper, we extend the considerations to prime-detecting quasi-modular forms of higher level, in particular describing the structure of the space of quasi-modular forms that detect primes in various arithmetic progressions. We also provide an ``analytic'' proof of the level one case.