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Abstract

We consider relations between the pairs of sequences $(f, g_f)$ generated by the Lambert series expansions $L_f(q) = \sum_{n \geq 1} f(n) q^n / (1-q^n)$ in $q$ where $g_f(m)$ is defined to be the coefficient of $q^m$ in $L_f(q)$. In particular, we prove new recurrence relations and matrix equations defining these sequences for all $n \in \mathbb{Z}^{+}$. The key ingredient to the proofs is Euler’s pentagonal number theorem. Our new results include new exact formulas for and applications to the Euler phi function $\phi(n)$, the Möbius function $\mu(n)$, the sum-of-divisors functions $\sigma_1(n)$ and $\sigma_{\alpha}(n)$ for $\alpha \geq 0$ and Liouville’s lambda function $\lambda(n)$.