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2024 | OriginalPaper | Chapter

The Partition Function Modulo 4

Author : Ken Ono

Published in: Class Groups of Number Fields and Related Topics

Publisher: Springer Nature Singapore

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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 generalized twisted Borcherds products developed by Bruinier and the author.

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previous chapter Connections of Class Numbers to the Group Structure of Generalized Pythagorean Triples

next chapter Triples, Quadruples and Quintuples Which are D(n)-Sets for Several n’s

We note that \(p(\alpha ):=0\) for non-integral \(\alpha \).

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Title: The Partition Function Modulo 4
Author: Ken Ono
Publisher: Springer Nature Singapore
Book: Class Groups of Number Fields and Related Topics
Print ISBN: 978-981-9769-10-0

Electronic ISBN: 978-981-9769-11-7

Copyright Year: 2024
DOI: https://doi.org/10.1007/978-981-97-6911-7_4

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