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For generalized Lambert series see Appell–Lerch sum.

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.

[edit] Examples

Since this last sum is a typical number-theoretic sum, almost any 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 $σ 0 (n) = d (n)$ 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 $α$ 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 $μ(n)$ :

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

For Euler's totient function $φ(n)$ :

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

For Liouville's function $λ(n)$ :

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

with the sum on the left similar to the Ramanujan theta function.

[edit] 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

$b_m = (a*1)(m) = \sum_{n\mid m} a_n\,$

as before. Examples of Lambert series in this form, with $z = 2π$ , occur in expressions for the Riemann zeta function for odd integer values; see Zeta constants for details.

[edit] Current usage

In the literature we find Lambert series applied to a wide variety of sums. For example, since $q n / (1 - q n) = Li 0 (q n)$ 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.

[edit] See also

Erdős–Borwein constant

[edit] References

Apostol, Tom M. (1976), Introduction to analytic number theory, Undergraduate Texts in Mathematics, New York-Heidelberg: Springer-Verlag, MR 0434929, ISBN 978-0-387-90163-3
Weisstein, Eric W., "Lambert Series" from MathWorld.

Contents

[edit] Examples

[edit] Alternate form

[edit] Current usage

[edit] See also

[edit] References