New recurrence relations and matrix equations for arithmetic functions generated by Lambert series

Maxie D. Schmidt¹

¹ School of Mathematics Georgia Institute of Technology Atlanta, GA 30332, U.S.A.

Acta Arithmetica, Tome 181 (2017) no. 4, pp. 355-367

Résumé

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 $\lambda(n)$.

Détail

DOI : 10.4064/aa170217-4-8

Keywords: consider relations between pairs sequences generated lambert series expansions sum geq q where defined coefficient particular prove recurrence relations matrix equations defining these sequences mathbb key ingredient proofs euler pentagonal number theorem results include exact formulas applications euler phi function phi bius function sum of divisors functions sigma sigma alpha alpha geq liouville lambda function nbsp lambda