Year: 2026
Type: preprint
Abstract: Recent developments in theories of partitions, quasimodular forms, multiple zeta values and $q$-multiple zeta values have led to connections between partitions and primes. Central to this progress are prime-detecting quasimodular forms, which are $q$-series whose non-negative coefficients vanish if and only if the input is prime. In this paper, we use properties of these quasimodular forms to motivate a proof of the infinitude of primes, to construct a novel partition theoretic factorization algorithm, and to show that the distribution of these coefficients relate to existence of primes in intervals. We also prove analogs of these results for primes in arithmetic progressions and discuss potential extensions of our results, including to Linnik's problem.
Source: arXiv (Cornell University)
Authors: Allen, Kevin, Craig, William
Institutions: University College Cork, United States Naval Academy
Language: English
Subfield: Algebra and Number Theory
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
Award: G8058152878
OpenAlex ID: W7214793756