Explicit transformations for generalized Lambert series associated with the divisor function ${\sigma }_a^{(N)}(n)$ and their applications

Soumyarup Banerjee, Atul Dixit, Shivajee Gupta

Let ${\sigma }_a^{(N)}(n)={\sum }_{d^N|n}{d}^a$. An explicit transformation is obtained for the generalized Lambert series ${\sum }_{n=1}^{\infty }{\sigma }_a^{(N)}(n)e^{-ny}$ for Re$(a)>-1$ using the…