Abstract

For a positive integer $t$, let $b_t(n)$ denote the number of $t$-regular partitions of a nonnegative integer $n$. In a recent paper, Keith and Zanello established infinite families of congruences and self-similarity results for $b_t(n)$ modulo $2$ for certain values of $t$. Further, they proposed some conjectures on self-similarities of $b_t(n)$ modulo $2$ for certain values of $t$. For example, for a positive proportion of primes $p$, they conjectured that $b_3(n)$ satisfies

where $\alpha \equiv -24^{-1}(\mathrm{mod}<!--FUNCTION\; APPLICATION-->p^2)$, $0<\alpha <p^2$. In this paper, we prove their conjectures on $b_3(n)$ and $b_{25}(n)$. We also prove a self-similarity result for $b_{21}(n)$ modulo $2$. The proofs use a result of Serre which says that if $f$ is an integer weight cusp form on the congruence subgroup ${\Gamma }_0(N)$, then for any positive integer $M$, a positive proportion of the primes $p\equiv -1(\mathrm{mod}<!--FUNCTION\; APPLICATION-->M“N)$ has the property that $f(z)\mid T_p\equiv 0(\mathrm{mod}<!--FUNCTION\; APPLICATION-->M)$.

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