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.
Related Lambert series over the Moebius function include the following identities for any
prime :
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:
The second identity in the previous equations follows from the fact that the coefficients of the left-hand-side sum are given by
where the function is the multiplicative identity with respect to the operation of Dirichlet convolution of arithmetic functions.
with the sum on the right similar to the Ramanujan theta function, or Jacobi theta function. Note that Lambert series in which the an are trigonometric functions, for example, an = sin(2nx), can be evaluated by various combinations of the logarithmic derivatives of Jacobi theta functions.
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
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[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
The next table lists the first several rows of these corresponding inverse matrices.[6]
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
2
1
1
1
0
0
0
0
5
4
3
2
1
1
0
0
0
6
5
3
2
2
1
1
0
0
7
10
7
5
3
2
1
1
0
8
12
9
6
4
3
2
1
1
We let denote the sequence of interleaved pentagonal numbers, i.e., so that the pentagonal number theorem is expanded in the form of
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
Recurrence relations[]
Within this section we define the following functions for natural numbers :
We also adopt the notation from the previous section that
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[]
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.
^Merca, Mircea (13 January 2017). "The Lambert series factorization theorem". The Ramanujan Journal. 44 (2): 417–435. doi:10.1007/s11139-016-9856-3.
^"A133732". Online Encyclopedia of Integer Sequences. Retrieved 22 April 2018.
^ abSchmidt, Maxie D. (8 December 2017). "New Recurrence Relations and Matrix Equations for Arithmetic Functions Generated by Lambert Series". Acta Arithmetica. 181 (4): 355–367. arXiv:1701.06257. Bibcode:2017arXiv170106257S. doi:10.4064/aa170217-4-8.
^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].
Berry, Michael V. (2010). Functions of Number Theory. CAMBRIDGE UNIVERSITY PRESS. pp. 637–641. ISBN978-0-521-19225-5.
Lambert, Preston A. (1904). "Expansions of algebraic functions at singular points". Proc. Am. Philos. Soc. 43 (176): 164–172. JSTOR983503.
Apostol, Tom M. (1976), Introduction to analytic number theory, Undergraduate Texts in Mathematics, New York-Heidelberg: Springer-Verlag, ISBN978-0-387-90163-3, MR0434929, Zbl0335.10001