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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 resummed 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.

Contents

ExamplesEdit

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 sigma functions, one has

 

where   is any complex number and

 

is the divisor function.

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 inverse with respect to the operation of Dirichlet convolution of arithmetic functions.

For Euler's totient 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 formEdit

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 usageEdit

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 theoremsEdit

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 relationsEdit

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]

 
 

DerivativesEdit

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 alsoEdit

ReferencesEdit

  1. ^ "Jupyter Notebook Viewer".
  2. ^ See the forum post here (or the article arXiv:1112.4911) and the conclusions section of this article 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. and Schmidt, M. D. (2018). "Generating Special Arithmetic Functions by Lambert Series Factorizations". Contributions to Discrete Mathematics. to appear. arXiv:1706.00393. Bibcode:2017arXiv170600393M.CS1 maint: multiple names: authors list (link)
  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: 355–367. arXiv:1701.06257. Bibcode:2017arXiv170106257S.
  8. ^ M. Merca and 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].