Since this last sum is a typical number-theoretic sum, almost any natural multiplicative function will be exactly summable when used in a Lambert series. Thus, for example, one has
where is the number of positive divisors of the number n.
The proof of the first identity above follows from a multi-section (or bisection) identity of these Lambert series generating functions in the following form where we denote to be the Lambert series generating function of the arithmetic function f:
Generally speaking, we can extend the previous generating function expansion by letting denote the characteristic function of the powers, , for positive natural numbers and defining the generalized m-Liouville lambda function to be the arithmetic function satisfying . This definition of clearly implies that , which in turn shows that
We also have a slightly more generalized Lambert series expansion generating the sum of squares function in the form of [3]
In 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
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.
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.
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.
A somewhat newer construction recently published over 2017–2018 relates to so-termed Lambert series factorization theorems of the form[4]
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.
n \ k
1
2
3
4
5
6
7
8
1
1
0
0
0
0
0
0
0
2
0
1
0
0
0
0
0
0
3
-1
-1
1
0
0
0
0
0
4
-1
0
-1
1
0
0
0
0
5
-1
-1
-1
-1
1
0
0
0
6
0
0
1
-1
-1
1
0
0
7
0
0
-1
0
-1
-1
1
0
8
1
0
0
1
0
-1
-1
1
Another characteristic form of the Lambert series factorization theorem expansions is given by[5]
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.[6]
Then for any Lambert series generating the sequence of , we have the corresponding inversion relation of the factorization theorem expanded above given by[7]
This work on Lambert series factorization theorems is extended in[8] to more general expansions of the form
where is any (partition-related) reciprocal generating function, is any arithmetic function, and where the modified coefficients are expanded by
The corresponding inverse matrices in the above expansion satisfy
so that as in the first variant of the Lambert factorization theorem above we obtain an inversion relation for the right-hand-side coefficients of the form
where is the infinite q-Pochhammer symbol. Then we have the following recurrence relations for involving these functions and the pentagonal numbers proved in:[7]
Derivatives of a Lambert series can be obtained by differentiation of the series termwise with respect to . We have the following identities for the termwise derivatives of a Lambert series for any [9][10]
where the bracketed triangular coefficients in the previous equations denote the Stirling numbers of the first and second kinds. We also have the next identity for extracting the individual coefficients of the terms implicit to the previous expansions given in the form of
Now if we define the functions for any by
where denotes Iverson's convention, then we have the coefficients for the derivatives of a Lambert series given by
Of course, by a typical argument purely by operations on formal power series we also have that
^See the forum post here (or the article arXiv:1112.4911) and the conclusions section of arXiv:1712.00611 by Merca and Schmidt (2018) for usage of these two less standard Lambert series for the Moebius function in practical applications.
^Weisstein, Eric W. "Lambert Series". MathWorld. Retrieved 22 April 2018.
^M. Merca & Schmidt, M. D. (2017). "New Factor Pairs for Factorizations of Lambert Series Generating Functions". arXiv:1706.02359 [math.CO].
^Schmidt, Maxie D. (2017). "Combinatorial Sums and Identities Involving Generalized Divisor Functions with Bounded Divisors". arXiv:1704.05595 [math.NT].
^Schmidt, Maxie D. (2017). "Factorization Theorems for Hadamard Products and Higher-Order Derivatives of Lambert Series Generating Functions". arXiv:1712.00608 [math.NT].