The partition function modulo 4

TL;DR

This paper proves the existence of infinitely many congruences modulo 4 among the partition function values, using modular forms, class numbers, and generalized twisted Borcherds products, challenging the belief that their parity is random.

Contribution

It constructs modular forms related to partition numbers and establishes infinite linear dependence congruences modulo 4, introducing new methods involving class numbers and Borcherds products.

Findings

Existence of infinitely many congruences modulo 4 among p(n) values.

Construction of modular forms congruent to twisted generating functions.

Development of new techniques using class numbers and Borcherds products.

Abstract

It is widely believed that the parity of the partition function $p(n)$ is ``random.'' Contrary to this expectation, in this note we prove the existence of infinitely many congruence relations modulo 4 among its values. For each square-free integer $1<D\equiv 23\pmod{24},$ we construct a weight 2 meromorphic modular form that is congruent modulo 4 to a certain twisted generating function for the numbers $p\big(\frac{Dm^2+1}{24}\big)\pmod 4$. We prove the existence of infinitely many linear dependence congruences modulo 4 among suitable sets of holomorphic normalizations of these series. These results rely on the theory of class numbers and Hilbert class polynomials, and {\it generalized twisted Borcherds products} developed by Bruinier and the author.