The Lambert series factorization theorem

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 Mobius function , Euler's totient , Jordan's totient , Liouville's function , the von Mangoldt function , and the divisor function . The fascinating feature of these identities is their common nature.