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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


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