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Lambert series

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Lambert series

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

Contents

S ( q ) = ∑ n = 1 ∞ a n q n 1 − q n .

It can be resummed formally by expanding the denominator:

S ( q ) = ∑ n = 1 ∞ a n ∑ k = 1 ∞ q n k = ∑ m = 1 ∞ b m q m

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

b m = ( a ∗ 1 ) ( m ) = ∑ n ∣ 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

∑ n = 1 ∞ q n σ 0 ( n ) = ∑ n = 1 ∞ 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

∑ n = 1 ∞ q n σ α ( n ) = ∑ n = 1 ∞ n α q n 1 − q n

where α is any complex number and

σ α ( n ) = ( Id α ∗ 1 ) ( n ) = ∑ d ∣ n d α

is the divisor function.

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.

Other Lambert series include those for the Möbius function μ ( n ) :

∑ n = 1 ∞ μ ( n ) q n 1 − q n = q .

For Euler's totient function φ ( n ) :

∑ n = 1 ∞ φ ( n ) q n 1 − q n = q ( 1 − q ) 2 .

For Liouville's function λ ( n ) :

∑ n = 1 ∞ λ ( n ) q n 1 − q n = ∑ n = 1 ∞ q n 2

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

Alternate form

Substituting q = e − z one obtains another common form for the series, as

∑ n = 1 ∞ a n e z n − 1 = ∑ m = 1 ∞ b m e − m z

where

b m = ( a ∗ 1 ) ( m ) = ∑ d ∣ m a d

as before. Examples of Lambert series in this form, with z = 2 π , 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 ) = L i 0 ( q n ) is a polylogarithm function, we may refer to any sum of the form

∑ n = 1 ∞ ξ n L i u ( α q n ) n s = ∑ n = 1 ∞ α n L i s ( ξ q n ) n u

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

12 ( ∑ n = 1 ∞ n 2 L i − 1 ( q n ) ) 2 = ∑ n = 1 ∞ n 2 L i − 5 ( q n ) − ∑ n = 1 ∞ n 4 L i − 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.

References

Lambert series Wikipedia
(Text) CC BY-SA


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