The Lambert series factorization theorem

Abstract

A factorization for partial sums of Lambert series is introduced in this paper. As corollaries, we derive some connections between partitions and divisors. These results can be easily used to discover and prove new combinatorial identities involving important functions from number theory: the Möbius function μ(n) , Euler’s totient φ(n) , Jordan’s totient Jk(n) , Liouville’s function λ(n) , the von Mangoldt function Λ (n) , and the divisor function σx(n). The fascinating feature of these identities is their common nature.