Lambert series and Liouville's identities
Volume 445 / 2007
Dissertationes Mathematicae 445 (2007), 1-72
MSC: 11R11, 11R27.
DOI: 10.4064/dm445-0-1
Abstract
The relationship between Liouville's arithmetic identities and products of Lambert series is investigated. For example it is shown that Liouville's arithmetic formula for the sum $$ \sum_{\textstyle {(a,b,x,y) \in \mathbb{N}^{{4}}\atop ax+by=n}} (F(a-b)-F(a+b)),$$ where $n\in \mathbb{N}$ and $F:\mathbb{Z} \rightarrow \mathbb{C}$ is an even function, is equivalent to the Lambert series for $$ \bigg( \sum_{n=1}^{\infty} \frac{q^n}{1-q^n} \sin n \theta \bigg)^2 \quad\ (\theta \in \mathbb{R} ,\, |q| <1)$$ given by Ramanujan.
