Summing Lambert Series in Euler's q-Exponential Functions

Ruiming Zhang

paper

Summing Lambert Series in Euler's q-Exponential Functions

arXiv:1811.07756

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 Möbius $μ (n)$ , the Euler's totient $φ (n)$ , Jordan's totient $J_{k} (n)$ , von Mangoldt $Λ (n)$ , divisor function $σ_{s} (n)$ , the Ramanujan's sum $c_{q} (n)$ , and sum of square functions $r_{2} (n), r_{4} (n), r_{8} (n)$ .