Lambert series

For generalized Lambert series, see Appell–Lerch sum.

Function $S(q)=\sum _{n=1}^{\infty }{\frac {q^{n}}{1-q^{n}}}$, represented as a Matplotlib plot, using a version of the Domain coloring method

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

$S(q)=\sum _{n=1}^{\infty }a_{n}{\frac {q^{n}}{1-q^{n}}}.$

It can be resummed formally by expanding the denominator:

$S(q)=\sum _{n=1}^{\infty }a_{n}\sum _{k=1}^{\infty }q^{nk}=\sum _{m=1}^{\infty }b_{m}q^{m}$

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

$b_{m}=(a*1)(m)=\sum _{n\mid m}a_{n}.\,$

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

Examples

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

$\sum _{n=1}^{\infty }q^{n}\sigma _{0}(n)=\sum _{n=1}^{\infty }{\frac {q^{n}}{1-q^{n}}}$

where $\sigma _{0}(n)=d(n)$ is the number of positive divisors of the number n.

For the higher order sigma functions, one has

$\sum _{n=1}^{\infty }q^{n}\sigma _{\alpha }(n)=\sum _{n=1}^{\infty }{\frac {n^{\alpha }q^{n}}{1-q^{n}}}$

where $\alpha$ is any complex number and

$\sigma _{\alpha }(n)=({\textrm {Id}}_{\alpha }*1)(n)=\sum _{d\mid n}d^{\alpha }\,$

is the divisor function.

Lambert series in which the a_n are trigonometric functions, for example, a_n = sin(2n x), can be evaluated by various combinations of the logarithmic derivatives of Jacobi theta functions.

Other Lambert series include those for the Möbius function $\mu (n)$:

$\sum _{n=1}^{\infty }\mu (n)\,{\frac {q^{n}}{1-q^{n}}}=q.$

For Euler's totient function $\varphi (n)$:

$\sum _{n=1}^{\infty }\varphi (n)\,{\frac {q^{n}}{1-q^{n}}}={\frac {q}{(1-q)^{2}}}.$

For Liouville's function $\lambda (n)$:

$\sum _{n=1}^{\infty }\lambda (n)\,{\frac {q^{n}}{1-q^{n}}}=\sum _{n=1}^{\infty }q^{n^{2}}$

with the sum on the right similar to the Ramanujan theta function, or Jacobi theta function $\vartheta _{3}(q)$.

We also have a slightly more generalized Lambert series expansion generating the sum of squares function $r_{2}(n)$ in the form of

$\sum _{n=1}^{\infty }{\frac {4\cdot (-1)^{n+1}q^{2n+1}}{1-q^{2n+1}}}=\sum _{m=1}^{\infty }r_{2}(m)q^{m}.$

In general, if we write the Lambert series over $f(n)$ which generates the arithmetic functions $g(m)=(f\ast 1)(m)$, the next pairs of functions correspond to other well-known convolutions expressed by their Lambert series generating functions in the forms of

$(f,g)=(\mu ,\varepsilon ),(\varphi ,\operatorname {Id} _{1}),(\lambda ,\chi _{\operatorname {sq} }),(\Lambda ,\log ),(|\mu |,2^{\omega }),(J_{t},\operatorname {Id} _{t}),(d^{3},(d\ast 1)^{2}),$

where $\varepsilon (n)=\delta _{n,1}$ is the multiplicative identity for Dirichlet convolutions, $\operatorname {Id} _{k}(n)=n^{k}$ is the identity function for $k^{th}$ powers, $\chi _{\operatorname {sq} }$ denotes the characteristic function for the squares, $\omega (n)$ which counts the number of distinct prime factors of $n$ (see Omega function), $J_{t}$ is Jordan's totient function, and $d(n)=\sigma _{0}(n)$ is the divisor function (see Dirichlet convolutions).

Alternate form

Substituting $q=e^{-z}$ one obtains another common form for the series, as

$\sum _{n=1}^{\infty }{\frac {a_{n}}{e^{zn}-1}}=\sum _{m=1}^{\infty }b_{m}e^{-mz}$

where

$b_{m}=(a*1)(m)=\sum _{d\mid m}a_{d}\,$

as before. Examples of Lambert series in this form, with $z=2\pi$, 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 $q^{n}/(1-q^{n})=\mathrm {Li} _{0}(q^{n})$ is a polylogarithm function, we may refer to any sum of the form

$\sum _{n=1}^{\infty }{\frac {\xi ^{n}\,\mathrm {Li} _{u}(\alpha q^{n})}{n^{s}}}=\sum _{n=1}^{\infty }{\frac {\alpha ^{n}\,\mathrm {Li} _{s}(\xi q^{n})}{n^{u}}}$

as a Lambert series, assuming that the parameters are suitably restricted. Thus

$12\left(\sum _{n=1}^{\infty }n^{2}\,\mathrm {Li} _{-1}(q^{n})\right)^{\!2}=\sum _{n=1}^{\infty }n^{2}\,\mathrm {Li} _{-5}(q^{n})-\sum _{n=1}^{\infty }n^{4}\,\mathrm {Li} _{-3}(q^{n}),$

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

$\sum _{n\geq 1}{\frac {a_{n}q^{n}}{1\pm q^{n}}}={\frac {1}{(\mp q;q)_{\infty }}}\sum _{n\geq 1}\left((s_{o}(n,k)\pm s_{e}(n,k))a_{k}\right)q^{n},$

where $s_{o}(n,k)\pm s_{e}(n,k)=[q^{n}](\mp q;q)_{\infty }{\frac {q^{k}}{1\pm q^{k}}}$ is the respective sum or difference of the restricted partition functions $s_{e/o}(n,k)$ which denote the number of $k$'s in all partitions of $n$ into an even (respectively, odd) number of distinct parts. Let $s_{n,k}:=s_{e}(n,k)-s_{o}(n,k)=[q^{n}](q;q)_{\infty }{\frac {q^{k}}{1-q^{k}}}$ 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

$L_{f}(q):=\sum _{n\geq 1}{\frac {f(n)q^{n}}{1-q^{n}}}={\frac {1}{(q;q)_{\infty }}}\sum _{n\geq 1}\left(s_{n,k}f(k)\right)q^{n},$

where $(q;q)_{\infty }$ 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

$s_{n,k}^{(-1)}=\sum _{d|n}p(d-k)\mu \left({\frac {n}{d}}\right)$

The next table lists the first several rows of these corresponding inverse matrices .

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 $G_{j}:={\frac {1}{2}}\left\lceil {\frac {j}{2}}\right\rceil \left\lceil {\frac {3j+1}{2}}\right\rceil$ denote the sequence of interleaved pentagonal numbers, i.e., so that the pentagonal number theorem is expanded in the form of

$(q;q)_{\infty }=\sum _{n\geq 0}(-1)^{\left\lceil {\frac {n}{2}}\right\rceil }q^{G_{n}}.$

Then for any Lambert series $L_{f}(q)$ generating the sequence of $g(n)=(f\ast 1)(n)$, we have the corresponding inversion relation of the factorization theorem expanded above given by

$f(n)=\sum _{k=1}^{n}\sum _{d|n}p(d-k)\mu (n/d)\times \sum _{j:k-G_{j}>0}(-1)^{\left\lceil {\frac {j}{2}}\right\rceil }b(k-G_{j}).$

This work on Lambert series factorization theorems is extended in to more general expansions of the form

$\sum _{n\geq 1}{\frac {a_{n}q^{n}}{1-q^{n}}}={\frac {1}{C(q)}}\sum _{n\geq 1}\left(\sum _{k=1}^{n}s_{n,k}(\gamma ){\widetilde {a}}_{k}(\gamma )\right)q^{n},$

where $C(q)$ is any (partition-related) reciprocal generating function, $\gamma (n)$ is any arithmetic function, and where the modified coefficients are expanded by

${\widetilde {a}}_{k}(\gamma )=\sum _{d|k}\sum _{r|{\frac {k}{d}}}a_{d}\gamma (r).$

The corresponding inverse matrices in the above expansion satisfy

$s_{n,k}^{(-1)}(\gamma )=\sum _{d|n}[q^{d-k}]{\frac {1}{C(q)}}\gamma \left({\frac {n}{d}}\right),$

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

${\widetilde {a}}_{k}(\gamma )=\sum _{k=1}^{n}s_{n,k}^{(-1)}(\gamma )\times [q^{k}]\left(\sum _{d=1}^{k}{\frac {a_{d}q^{q}}{1-q^{d}}}C(q)\right).$

See also

• Erdős–Borwein constant

References

• Berry, Michael V. (2010). Functions of Number Theory. CAMBRIDGE UNIVERSITY PRESS. pp. 637–641. ISBN 978-0-521-19225-5.
• Lambert, Preston A. (1904). "Expansions of algebraic functions at singular points". Proc. Am. Philos. Soc. 43 (176): 164–172. JSTOR 983503.
• Apostol, Tom M. (1976), Introduction to analytic number theory, Undergraduate Texts in Mathematics, New York-Heidelberg: Springer-Verlag, ISBN 978-0-387-90163-3, MR 0434929, Zbl 0335.10001
• "Lambert series", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
• Weisstein, Eric W. "Lambert Series". MathWorld.