Lambert series

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

It can be resummed formally by expanding the denominator:

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

This series may be inverted by means of the Möbius inversion formula, and is an example of a Möbius transform.

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

where ${\displaystyle \sigma _{0}(n)=d(n)}$ is the number of positive divisors of the number n.

For the higher order sigma functions, one has

where ${\displaystyle \alpha }$ is any complex number and

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