Voronoi summation formulas for the product of the generalized divisor function and the Liouville lambda function

The Voronoi summation formulas for the divisor function and $r_2(n)$ are well known. Not only do these formulas have interesting structure, but they have also been used to improve the error term in the Dirichlet divisor problem and the Gauss circle problem, respectively. In this work with Atul Dixit, we derive Voronoi summation formulas for some other functions related to the generalized divisor function and the Liouville lambda function. We also derive beautiful analogues of Cohen’s identity and the Ramanujan–Guinand formula associated to these functions and show their equivalence.