Arithmetic of the 13-regular partition function modulo 3 (Q539132)

In this article, the author studies the arithmetic function \(b_{13}(n)\) given by \[ \sum_{n=0}^\infty b_{13}(n)q^n = \prod_{m=1}^\infty \frac{(1-q^{13m})}{(1-q^m)}. \] Using the theory of modular forms, in particular, Theorems of Sturm and Newman, the author identifies \[ \sum_{n=0}^\infty b_{13}(3n+1)q^{6n+3} \pmod{3} \] with a weight 12 modular form \(H(z)\) of level 312. The author then establishes for example the congruence \[ H(z)|T_3\equiv -H(z)|T_3^2\pmod{3} \] and deduces that \[ b_{13}(3n+1)\equiv -b_{13}(9n+4)\pmod{3}. \] The author also proves that \[ b_{13}(3n+1) \equiv (-1)^{\ell+1}b_{13}\left(3^\ell n+\frac{3^\ell-1}{2}\right)\pmod{3}. \]