It can be resummed formally by expanding the denominator:
where the coefficients of the new series are given by the Dirichlet convolution of with the constant function :
where is the number of positive divisors of the number .
For the higher order sigma functions, one has
Other Lambert series include those for the Möbius function :
For Euler's totient function :
For Liouville's function :
with the sum on the left similar to the Ramanujan theta function.
In the literature we find Lambert series applied to a wide variety of sums. For example, since is a polylogarithm function, we may refer to any sum of the form
as a Lambert series, assuming that the parameters are suitably restricted. Thus
which holds for all complex 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.
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