general, if we write the Lambert series over which generates the arithmetic functions , the next pairs of functions correspond to other well-known convolutions expressed by their Lambert series generating functions in the forms of where is the multiplicative identity for Dirichlet convolutions, is the identity function for powers, denotes the characteristic function for the squares, which counts the number of distinct prime factors of (see prime omega function), is Jordan's totient function, and is the divisor function (see Dirichlet convolutions). The conventional use of the letter q in the summations is a historical usage, referring to its origins in the theory of elliptic curves and theta functions, as the nome. Alternate form Substituting one obtains another common form for the series, as where as before. Examples of Lambert series in this form, with , occur in expressions for the Riemann zeta function for odd integer values; see Zeta constants for details. Current usage In the literature we find Lambert series applied to a wide variety of sums. For example, since is a polylogarithm function, we may refer to any sum of the form as a Lambert series, assuming that the parameters are suitably restricted. Thus which holds for all complex q not on the unit circle, would be considered a Lambert series identity. This identity follows in a straightforward fashion from some identities published by the Indian mathematician S. Ramanujan. A very thorough exploration of Ramanujan's works can be found in the works by Bruce Berndt. Factorization theorems A somewhat newer construction recently published over 2017–2018 relates to so-termed Lambert series factorization theorems of the form where is the respective sum or difference of the restricted partition functions which denote the number of 's in all partitions of into an even (respectively, odd) number of distinct parts. Let denote the invertible lower triangular sequence whose first few values are shown in the table below. Another characteristic form of the Lambert series factorization theorem expansions is given by where is the (infinite) q-Pochhammer symbol. The invertible matrix products on the right-hand-side of the previous equation correspond to inverse matrix products whose lower triangular entries are given in terms of the partition function and the Möbius function by the divisor sums The next table lists the first several rows of these corresponding inverse matrices. We let denote the sequence of interleaved pentagonal numbers, i.e., so that the pentagonal number theorem is