Lambert series

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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) =	∑	a_n
	n \| m

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

1 Examples
2 Alternate form
3 See also
4 References

[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

σ_α(n) = (Id_α * 1)(n) =	∑	d^α
	d \| n

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 Mobius function $μ(n)$ :

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

For Euler's totient function $φ(n)$ :

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

For Liouville's function $λ(n)$ :

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

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) =	∑	a_n
	n \| m

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] See also

Erdős–Borwein constant

[edit] References

Tom M. Apostol, Introduction to Analytic Number Theory, (1976) Springer-Verlag, New York. ISBN 0-387-90163-9
Eric W. Weisstein, Lambert Series at MathWorld.

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Categories: Analytic number theory | Q-analogs | Mathematical series