Quasimodular Forms, Partitions, and Primes in Intervals

Source 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.