math lessons - 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 a n n | m

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) = \sum 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.

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