Lambert series

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

<math>S(q)=\sum_{n=1}^\infty a_n \frac {q^n}{1-q^n}<math>

It can be resummed formally by expanding the denominator:

<math>S(q)=\sum_{n=1}^\infty a_n \sum_{k=1}^\infty q^{nk} = \sum_{m=1}^\infty b_m q^m <math>

where the coefficients of the new series are given by the Dirichlet convolution of <math>{a_n}<math> with the constant function <math>1(n)=1<math>:

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

<math>\sum_{n=1}^{\infty} q^n \sigma_0(n) = \sum_{n=1}^{\infty} \frac{q^n}{1-q^n}<math>

where <math>\sigma_0(n)=d(n)<math> is the number of positive divisors of the number <math>n<math>.

For the higher order sigma functions, one has

<math>\sum_{n=1}^{\infty} q^n \sigma_\alpha(n) = \sum_{n=1}^{\infty} \frac{n^\alpha q^n}{1-q^n}<math>

where <math>\alpha<math> is any complex number and

<math>\sigma_\alpha(n) = (\textrm{Id}_\alpha*1)(n) = \sum_{d|n} d^\alpha<math>

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.

Lambert series

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