Lambert series#Factorization theorems

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

:$\backslash sum\_\{n=1\}^\backslash infty\; q^n\; \backslash sigma\_0(n)\; =\; \backslash sum\_\{n=1\}^\backslash infty\; \backslash frac\{q^n\}\{1-q^n\}$

where $\backslash sigma\_0(n)=d(n)$ is the number of positive divisors of the number n.

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

:$\backslash sum\_\{n=1\}^\backslash infty\; q^n\; \backslash sigma\_\backslash alpha(n)\; =\; \backslash sum\_\{n=1\}^\backslash infty\; \backslash frac\{n^\backslash alpha\; q^n\}\{1-q^n\}\; =\; \backslash sum\_\{n=1\}^\backslash infty\; \backslash operatorname\{Li\}\_\{-\backslash alpha\}(q^n)$

where $\backslash alpha$ is any complex number, $\backslash operatorname\{Li\}$ is the polylogarithm, and

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

is the divisor function. In particular, for $\backslash alpha\; =\; 1$, the Lambert series one gets is

:$q\; \backslash frac\{F\text{'}(q)\}\{F(q)\}$

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

:$F(q)\; :=\; \backslash frac\{1\}\{\backslash phi(q)\}\; =\; \backslash sum\_\{k=0\}^\backslash infty\; p(k)\; q^k\; =\; \backslash prod\_\{n=1\}^\backslash infty\; \backslash frac\{1\}\{1-q^n\}.$

Additional Lambert series related to the previous identity include those for the variants of the

Möbius function given below $\backslash mu(n)$

:See the forum post [https://mathoverflow.net/q/98174 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.

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

Related Lambert series over the Möbius function include the following identities for any

prime $\backslash alpha\; \backslash in\; \backslash mathbb\{Z\}^\{+\}$:

:$$

\sum_{n \geq 1} \frac{\mu(n) q^n}{1+q^n} = q-2q^2

:$$

\sum_{n \geq 1} \frac{\mu(\alpha n) q^n}{1-q^n} = -\sum_{n \geq 0} q^{\alpha^n}.

{{Citation needed|date=May 2024}}

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

$L\_\{f\}(q)\; :=\; q$ to be the Lambert series generating function of the arithmetic function f:

:$$

\begin{align}

\sum_{n \geq 1} \frac{f(n) q^n}{1+q^n} & = \sum_{n \geq 1} \frac{f(n)q^{n}}{1-q^{n}} -

\sum_{n \geq 1} \frac{2 f(n) q^{2n}}{1-q^{2n}} \\

& =

L_f(q) - 2 \cdot L_f(q^2).

\end{align}

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

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

For Von Mangoldt function $\backslash Lambda(n)$:

:$\backslash sum\_\{n=1\}^\backslash infty\; \backslash Lambda(n)\backslash ,\backslash frac\{q^n\}\{1-q^n\}\; =\; \backslash sum\_\{n=1\}^\{\backslash infty\}\; \backslash log(n)q^n$

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

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

\sum_{n=1}^\infty q^{n^2}

with the sum on the right similar to the Ramanujan theta function, or Jacobi theta function $\backslash vartheta\_3(q)$. Note that 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.

Generally speaking, we can extend the previous generating function expansion by letting $\backslash chi\_m(n)$ denote the characteristic function of the $m^\{th\}$ powers, $n\; =\; k^m\; \backslash in\; \backslash mathbb\{Z\}^\{+\}$, for positive natural numbers $m\; >\; 2$ and defining the generalized m-Liouville lambda function to be the arithmetic function satisfying $\backslash chi\_m(n)\; :=\; (1\; \backslash ast\; \backslash lambda\_m)(n)$. This definition of $\backslash lambda\_m(n)$ clearly implies that $\backslash lambda\_m(n)\; =\; \backslash sum\_\{d^m|n\}\; \backslash mu\backslash left(\backslash frac\{n\}\{d^m\}\backslash right)$, which in turn shows that

:$\backslash sum\_\{n\; \backslash geq\; 1\}\; \backslash frac\{\backslash lambda\_m(n)\; q^n\}\{1-q^n\}\; =\; \backslash sum\_\{n\; \backslash geq\; 1\}\; q^\{n^m\},\backslash \; \backslash text\{\; for\; \}\; m\; \backslash geq\; 2.$

We also have a slightly more generalized Lambert series expansion generating the sum of squares function $r\_2(n)$ in the form of

{{cite web|last1=Weisstein|first1=Eric W.|title=Lambert Series|url=http://mathworld.wolfram.com/LambertSeries.html|website=MathWorld|access-date=22 April 2018}}

:$\backslash sum\_\{n=1\}^\{\backslash infty\}\; \backslash frac\{4\; \backslash cdot\; (-1)^\{n+1\}\; q^\{2n+1\}\}\{1-q^\{2n+1\}\}\; =\; \backslash sum\_\{m=1\}^\{\backslash infty\}\; r\_2(m)\; q^m.$

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

:$(f,\; g)\; =\; (\backslash mu,\; \backslash varepsilon),\; (\backslash varphi,\; \backslash operatorname\{Id\}\_1),\; (\backslash lambda,\; \backslash chi\_\{\backslash operatorname\{sq\}\}),\; (\backslash Lambda,\; \backslash log),$

(|\mu|, 2^{\omega}), (J_t, \operatorname{Id}_t), (d^3, (d \ast 1)^2),

where $\backslash varepsilon(n)\; =\; \backslash delta\_\{n,1\}$ is the multiplicative identity for Dirichlet convolutions, $\backslash operatorname\{Id\}\_k(n)\; =\; n^k$ is the identity function for $k^\{th\}$ powers, $\backslash chi\_\{\backslash operatorname\{sq\}\}$ denotes the characteristic function for the squares, $\backslash omega(n)$ which counts the number of distinct prime factors of $n$ (see prime omega function), $J\_t$ is Jordan's totient function, and $d(n)\; =\; \backslash sigma\_0(n)$ 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.

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

:$\backslash sum\_\{n=1\}^\backslash infty\; \backslash frac\; \{a\_n\}\{e^\{zn\}-1\}=\; \backslash sum\_\{m=1\}^\backslash infty\; b\_m\; e^\{-mz\}$

where

:$b\_m\; =\; (a*1)(m)\; =\; \backslash sum\_\{d\backslash mid\; m\}\; a\_d\backslash ,$

as before. Examples of Lambert series in this form, with $z=2\backslash pi$, 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 $q^n/(1\; -\; q^n\; )\; =\; \backslash mathrm\{Li\}\_0(q^\{n\})$ is a polylogarithm function, we may refer to any sum of the form

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

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

:$12\backslash left(\backslash sum\_\{n=1\}^\{\backslash infty\}\; n^2\; \backslash ,\; \backslash mathrm\{Li\}\_\{-1\}(q^n)\backslash right)^\{\backslash !2\}\; =\; \backslash sum\_\{n=1\}^\{\backslash infty\}$

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.

A somewhat newer construction recently published over 2017–2018 relates to so-termed Lambert series factorization theorems of the form{{cite journal|last1=Merca|first1=Mircea|title=The Lambert series factorization theorem|journal=The Ramanujan Journal|date=13 January 2017|volume=44|issue=2|pages=417–435|doi=10.1007/s11139-016-9856-3|s2cid=125286799}}

:$\backslash sum\_\{n\; \backslash geq\; 1\}\; \backslash frac\{a\_n\; q^n\}\{1\backslash pm\; q^n\}\; =\; \backslash frac\{1\}\{(\backslash mp\; q;\; q)\_\{\backslash infty\}\}\; \backslash sum\_\{n\; \backslash geq\; 1\}\; \backslash left((s\_o(n,\; k)\; \backslash pm\; s\_e(n,\; k))\; a\_k\backslash right)\; q^n,$

where $s\_o(n,\; k)\; \backslash pm\; s\_e(n,\; k)\; =\; [q^n]\; (\backslash mp\; q;\; q)\_\{\backslash infty\}\; \backslash frac\{q^k\}\{1\; \backslash pm\; q^k\}$ 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 $s\_\{n,k\}\; :=\; s\_e(n,\; k)\; -\; s\_o(n,\; k)\; =\; [q^n]\; (q;\; q)\_\{\backslash infty\}\; \backslash frac\{q^k\}\{1-q^k\}$ denote the invertible lower triangular sequence whose first few values are shown in the table below.

Another characteristic form of the Lambert series factorization theorem expansions is given by{{cite journal|author=Merca, M.|author2=Schmidt, M. D.|name-list-style=amp|title=Generating Special Arithmetic Functions by Lambert Series Factorizations|journal=Contributions to Discrete Mathematics|date=2019|volume=14|issue=1|pages=31–45|doi=10.11575/cdm.v14i1.62425|doi-access=free|arxiv=1706.00393|bibcode=2017arXiv170600393M}}

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

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

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

The next table lists the first several rows of these corresponding inverse matrices.{{cite web|title=A133732|url=https://oeis.org/A133732|website=Online Encyclopedia of Integer Sequences|access-date=22 April 2018}}

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

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

Then for any Lambert series $L\_f(q)$ generating the sequence of $g(n)\; =\; (f\; \backslash ast\; 1)(n)$, we have the corresponding inversion relation of the factorization theorem expanded above given by{{cite journal|last1=Schmidt|first1=Maxie D.|title=New Recurrence Relations and Matrix Equations for Arithmetic Functions Generated by Lambert Series|journal=Acta Arithmetica|date=8 December 2017|volume=181|issue=4|pages=355–367|doi=10.4064/aa170217-4-8|arxiv=1701.06257|bibcode=2017arXiv170106257S|s2cid=119130467}}

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

This work on Lambert series factorization theorems is extended in{{cite arXiv|author=M. Merca|author2=Schmidt, M. D.|name-list-style=amp|title=New Factor Pairs for Factorizations of Lambert Series Generating Functions|eprint=1706.02359|class=math.CO|year=2017}} to more general expansions of the form

:$\backslash sum\_\{n\; \backslash geq\; 1\}\; \backslash frac\{a\_n\; q^n\}\{1-q^n\}\; =\; \backslash frac\{1\}\{C(q)\}\; \backslash sum\_\{n\; \backslash geq\; 1\}\; \backslash left(\backslash sum\_\{k=1\}^n\; s\_\{n,k\}(\backslash gamma)\; \backslash widetilde\{a\}\_k(\backslash gamma)\backslash right)\; q^n,$

where $C(q)$ is any (partition-related) reciprocal generating function, $\backslash gamma(n)$ is any arithmetic function, and where the

modified coefficients are expanded by

:$\backslash widetilde\{a\}\_k(\backslash gamma)\; =\; \backslash sum\_\{d|k\}\; \backslash sum\_\{r|\; \backslash frac\{k\}\{d\}\}\; a\_d\; \backslash gamma(r).$

The corresponding inverse matrices in the above expansion satisfy

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

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

:$\backslash widetilde\{a\}\_k(\backslash gamma)\; =\; \backslash sum\_\{k=1\}^\{n\}\; s\_\{n,k\}^\{(-1)\}(\backslash gamma)\; \backslash times\; [q^k]\backslash left(\backslash sum\_\{d=1\}^k\; \backslash frac\{a\_d\; q^d\}\{1-q^d\}\; C(q)\backslash right).$

Within this section we define the following functions for natural numbers $n,x\; \backslash geq\; 1$:

:$g\_f(n)\; :=\; (f\; \backslash ast\; 1)(n),$

:$\backslash Sigma\_f(x)\; :=\; \backslash sum\_\{1\; \backslash leq\; n\; \backslash leq\; x\}\; g\_f(n).$

We also adopt the notation from the previous section that

:$s\_\{n,k\}\; =\; [q^n]\; (q;\; q)\_\{\backslash infty\}\; \backslash frac\{q^k\}\{1-q^k\},$

where $(q;\; q)\_\{\backslash infty\}$ is the infinite q-Pochhammer symbol. Then we have the following recurrence relations for involving these functions and the pentagonal numbers proved in:

:$g\_f(n+1)\; =\; \backslash sum\_\{b\; =\; \backslash pm\; 1\}\; \backslash sum\_\{k=1\}^\{\backslash left\backslash lfloor\; \backslash frac\{\backslash sqrt\{24n+1\}-b\}\{6\}\backslash right\backslash rfloor\}$

(-1)^{k+1} g_f\left(n+1-\frac{k(3k+b)}{2}\right) +

\sum_{k=1}^{n+1} s_{n+1,k} f(k),

:$\backslash Sigma\_f(x+1)\; =\; \backslash sum\_\{b\; =\; \backslash pm\; 1\}\; \backslash sum\_\{k=1\}^\{\backslash left\backslash lfloor\; \backslash frac\{\backslash sqrt\{24x+1\}-b\}\{6\}\backslash right\backslash rfloor\}$

(-1)^{k+1} \Sigma_f\left(n+1-\frac{k(3k+b)}{2}\right) +

\sum_{n=0}^x \sum_{k=1}^{n+1} s_{n+1,k} f(k).

Derivatives of a Lambert series can be obtained by differentiation of the series termwise with respect to $q$. We have the following identities for the termwise $s^\{th\}$ derivatives of a Lambert series for any $s\; \backslash geq\; 1${{cite arXiv|last1=Schmidt|first1=Maxie D.|title=Combinatorial Sums and Identities Involving Generalized Divisor Functions with Bounded Divisors|date=2017|eprint=1704.05595|class=math.NT}}{{cite arXiv|last1=Schmidt|first1=Maxie D.|title=Factorization Theorems for Hadamard Products and Higher-Order Derivatives of Lambert Series Generating Functions|date=2017|eprint=1712.00608|class=math.NT}}

:$q^s\; \backslash cdot\; D^\{(s)\}\backslash left[\backslash frac\{q^i\}\{1-q^i\}\backslash right]\; =\; \backslash sum\_\{m=0\}^s\; \backslash sum\_\{k=0\}^m\; \backslash left[\backslash begin\{matrix\}\; s\; \backslash \backslash \; m\backslash end\{matrix\}\backslash right]$

\left\{\begin{matrix} m \\ k\end{matrix}\right\} \frac{(-1)^{s-k} k! i^m}{(1-q^i)^{k+1}}

:$q^s\; \backslash cdot\; D^\{(s)\}\backslash left[\backslash frac\{q^i\}\{1-q^i\}\backslash right]\; =\; \backslash sum\_\{r=0\}^s\backslash left[\backslash sum\_\{m=0\}^s\; \backslash sum\_\{k=0\}^m\; \backslash left[\backslash begin\{matrix\}\; s\; \backslash \backslash \; m\backslash end\{matrix\}\backslash right]$

\left\{\begin{matrix} m \\ k\end{matrix}\right\} \binom{s-k}{r} \frac{(-1)^{s-k-r} k! i^m}{(1-q^i)^{k+1}}\right] q^{(r+1)i},

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

:$[q^n]\backslash left(\backslash sum\_\{i\; \backslash geq\; t\}\; \backslash frac\{a\_i\; q^\{mi\}\}\{(1-q^i)^\{k+1\}\}\backslash right)\; =\; \backslash sum\_\{\backslash begin\{matrix\}\; d|n\; \backslash \backslash \; t\; \backslash leq\; d\; \backslash leq\; \backslash left\backslash lfloor\; \backslash frac\{n\}\{m\}\; \backslash right\backslash rfloor\backslash end\{matrix\}\}$

\binom{\frac{n}{d}-m+k}{k} a_d.

Now if we define the functions $A\_t(n)$ for any $n,t\; \backslash geq\; 1$ by

:$A\_t(n)\; :=\; \backslash sum\_\{\backslash begin\{matrix\}\; 0\; \backslash leq\; k\; \backslash leq\; m\; \backslash leq\; t\; \backslash \backslash \; 0\; \backslash leq\; r\; \backslash leq\; t\backslash end\{matrix\}\}\; \backslash sum\_\{d|n\}\; \backslash left[\backslash begin\{matrix\}\; t\; \backslash \backslash \; m\backslash end\{matrix\}\backslash right]$

\left\{\begin{matrix} m \\ k\end{matrix}\right\} \binom{t-k}{r} \binom{\frac{n}{d}-1-r+k}{k} (-1)^{t-k-r} k! d^m \cdot a_d \cdot

\left[t \leq d \leq \left\lfloor \frac{n}{r+1} \right\rfloor\right]_{\delta},

where $[\backslash cdot]\_\{\backslash delta\}$ denotes Iverson's convention, then we have the coefficients for the $t^\{th\}$ derivatives of a Lambert series

given by

:$\backslash begin\{align\}$

A_t(n) & = [q^n]\left(q^t \cdot D^{(t)}\left[\sum_{i \geq t} \frac{a_i q^i}{1-q^i}\right]\right) \\

& = [q^n]\left(\sum_{n \geq 1} \frac{(A_t \ast \mu)(n) q^n}{1-q^n}\right).

\end{align}

Of course, by a typical argument purely by operations on formal power series we also have that

:$[q^n]\backslash left(q^t\; \backslash cdot\; D^\{(t)\}\backslash left[\backslash sum\_\{i\; \backslash geq\; 1\}\; \backslash frac\{f(i)\; q^i\}\{1-q^i\}\backslash right]\backslash right)\; =\; \backslash frac\{n!\}\{(n-t)!\}\; \backslash cdot\; (f\; \backslash ast\; 1)(n).$

• Erdős–Borwein constant
• Arithmetic function
• Dirichlet convolution

{{Reflist}}

• {{cite book|last=Berry|first=Michael V. |title=Functions of Number Theory|year=2010|publisher=CAMBRIDGE UNIVERSITY PRESS|isbn=978-0-521-19225-5|pages=637–641|url=http://dlmf.nist.gov}}
• {{cite journal|first1=Preston A. | last1=Lambert

|title=Expansions of algebraic functions at singular points

|journal=Proc. Am. Philos. Soc.

|year=1904

|volume=43 | number=176

|jstor=983503

|pages=164–172

}}

• {{Apostol IANT}}
• {{springer|title=Lambert series|id=p/l057340}}
• {{MathWorld|urlname=LambertSeries|title=Lambert Series}}
• {{cite arXiv|last=Schmidt|first=Maxie Dion|date=2020-04-06|title=A catalog of interesting and useful Lambert series identities|class=math.NT|eprint=2004.02976}}

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