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$23$-REGULAR PARTITIONS AND MODULAR FORMS WITH COMPLEX MULTIPLICATION
Published online by Cambridge University Press: 23 December 2022
Abstract
A partition of a positive integer n is called $\ell $-regular if none of its parts is divisible by $\ell $. Denote by $b_{\ell }(n)$ the number of $\ell $-regular partitions of n. We give a complete characterisation of the arithmetic of $b_{23}(n)$ modulo $11$ for all n not divisible by $11$ in terms of binary quadratic forms. Our result is obtained by establishing a relation between the generating function for these values of $b_{23}(n)$ and certain modular forms having complex multiplication by ${\mathbb Q}(\sqrt {-69})$.
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- Type
- Research Article
- Information
- Bulletin of the Australian Mathematical Society , Volume 108 , Issue 2 , October 2023 , pp. 254 - 263
- Copyright
- © The Author(s), 2022. Published by Cambridge University Press on behalf of Australian Mathematical Publishing Association Inc.