Jacobi forms and a certain space of modular forms.
Let λ(n) be the Liouville function. We find a nontrivial upper bound for the sum The main tool we use is Vaughan’s identity for λ(n).
Let be the Ramanujan sum, i.e. , where μ is the Möbius function. In a paper of Chan and Kumchev (2012), asymptotic formulas for (k = 1,2) are obtained. As an analogous problem, we evaluate (k = 1,2), where .