Advanced Search

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 ${\sum }_{\genfrac{}{}{0pt}{}{(a,b,x,y)\in \mathbb{N}⁴}{ax+by=n}}(F(a-b)-F(a+b))$, where n ∈ ℕ and F: ℤ → ℂ is an even function, is equivalent to the Lambert series for $\left({\sum }_{n=1}^{\infty }(qⁿ/(1-qⁿ))sinn\theta \right)²$ (θ ∈ ℝ, |q| < 1) given by Ramanujan.