Congruences in fractional partition functions (1908.03937v3)

Abstract: The coefficients of the generating function $(q;q)^\alpha_\infty$ produce $p_\alpha(n)$ for $\alpha \in \mathbb{Q}$. In particular, when $\alpha = -1$, the partition function is obtained. Recently, Chan and Wang identified and proved congruences of the form $p_{\frac{a}{b}}(\ell n + c)\equiv 0 \pmod{\ell}$ where $\ell$ is a prime such that $\ell \mid a -db$ for $d \in {4, 6, 8, 10, 14, 26}$. Expanding upon their work, we use the representation of powers of the Dedekind-eta functions in linear sums of Hecke eigenforms and their lacunarity to raise the power of the modulus to higher powers of $\ell$. In addition, we generate congruences for when $d=2$ employing Hecke algebra.