Sigma functions

Lambert series

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_{nmid 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 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 $sigma_0(n)=d(n)$ 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

alpha

is any complex number and

sigma_alpha(n) = (textrm{Id}_alpha*1)(n) = sum_{dmid 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 $mu(n)$ :

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

For Euler's totient function $phi(n)$ :

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

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

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.

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_{nmid m} a_n,

as before. Examples of Lambert series in this form, with $z=2pi$ , 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 $q^n/(1 - q^n ) = mathrm{Li}_0(q^{n})$ 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

12left(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.