Lambert series

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Function , represented as a Matplotlib plot, using a version of the Domain coloring method[1]

In mathematics, a Lambert series, named for Johann Heinrich Lambert, is a series taking the form

It can be resumed formally by expanding the denominator:

where the coefficients of the new series are given by the Dirichlet convolution of an with the constant function 1(n) = 1:

This series may be inverted by means of the Möbius inversion formula, and is an example of a Möbius transform.

Examples[edit]

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.

For the higher order sum-of-divisor functions, one has

where is any complex number and

is the divisor function. In particular, for , the Lambert series one gets is

which is (up to the factor of ) the logarithmic derivative of the usual generating function for partition numbers

Additional Lambert series related to the previous identity include those for the variants of the Möbius function given below

[2]

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.

For Euler's totient function :

For Von Mangoldt function :

For Liouville's function :

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(2n x), 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[edit]

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[edit]

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[edit]

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]

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[edit]

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[edit]

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 also[edit]

References[edit]

  1. ^ "Jupyter Notebook Viewer".
  2. ^ 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.
  3. ^ Weisstein, Eric W. "Lambert Series". MathWorld. Retrieved 22 April 2018.
  4. ^ Merca, Mircea (13 January 2017). "The Lambert series factorization theorem". The Ramanujan Journal. 44 (2): 417–435. doi:10.1007/s11139-016-9856-3.
  5. ^ Merca, M. & Schmidt, M. D. (2019). "Generating Special Arithmetic Functions by Lambert Series Factorizations". Contributions to Discrete Mathematics. 14 (1): 31–45. arXiv:1706.00393. Bibcode:2017arXiv170600393M. doi:10.11575/cdm.v14i1.62425.
  6. ^ "A133732". Online Encyclopedia of Integer Sequences. Retrieved 22 April 2018.
  7. ^ a b Schmidt, 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.
  8. ^ M. Merca & Schmidt, M. D. (2017). "New Factor Pairs for Factorizations of Lambert Series Generating Functions". arXiv:1706.02359 [math.CO].
  9. ^ Schmidt, Maxie D. (2017). "Combinatorial Sums and Identities Involving Generalized Divisor Functions with Bounded Divisors". arXiv:1704.05595 [math.NT].
  10. ^ Schmidt, Maxie D. (2017). "Factorization Theorems for Hadamard Products and Higher-Order Derivatives of Lambert Series Generating Functions". arXiv:1712.00608 [math.NT].