Summing Lambert Series in Euler's q-Exponential Functions

Abstract

In the work we shall present formulas to sum Lambert series using Euler's q-exponential functions, and several Lambert series associated with well-known arithmetic functions are given as examples. These functions are: the M\"{o}bius $\mu(n)$, the Euler's totient $\varphi(n)$, Jordan's totient $J_{k}(n)$, von Mangoldt $\Lambda(n)$, divisor function $\sigma_{s}(n)$, the Ramanujan's sum $c_{q}(n)$ , and sum of square functions $r_{2}(n),r_{4}(n),r_{8}(n)$.