Liouville Function - Series

Series

The Dirichlet series for the Liouville function gives the Riemann zeta function as

The Lambert series for the Liouville function is

\sum_{n=1}^\infty \frac{\lambda(n)q^n}{1-q^n} = \sum_{n=1}^\infty q^{n^2} = \frac{1}{2}\left(\vartheta_3(q)-1\right),

where is the Jacobi theta function.

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