Lambert Series

A series of the form

$$F(x)\equiv \sum_{n=1}^\infty a_n {x^n\over 1-x^n}$$ (1)

for $\vert x\vert<1$. Then

$$F(x)=\sum_{n=1}^\infty a_n \sum_{m=1}^\infty x^{mn}=\sum_{N=1}^\infty b_N x^N,$$ (2)

where

$$b_N\equiv \sum_{n\vert N} a_n.$$ (3)

Some beautiful series of this type include

$$\sum_{n=1}^\infty {\mu(n)x^n\over 1-x^n} = x$$ (4)

$$\sum_{n=1}^\infty {\phi(n)x^n\over 1-x^n} = {x\over(1-x)^2}$$ (5)

$$\sum_{n=1}^\infty {x^n\over 1-x^n} = \sum_{n=1}^\infty d(n)x^n$$ (6)

$$\sum_{n=1}^\infty {n^k x^n\over 1-x^n} = \sum_{n=1}^\infty \sigma_k(n)x^n$$ (7)

$$\sum_{n=1}^\infty {4(-1)^{n+1}x^n\over 1-x^n} = \sum_{n=1}^\infty r(n)x^n,$$ (8)

where $\mu(n)$ is the Möbius Function, $\phi(n)$ is the Totient Function, $d(n)=\sigma_0(n)$ is the number of divisors of $n$, $\sigma_k(n)$ is the Divisor Function, and $r(n)$ is the number of representations of $n$ in the form $n=A^2+B^2$ where $A$ and $B$ are rational integers (Hardy and Wright 1979).

References

Abramowitz, M. and Stegun, C. A. (Eds.). ``Number Theoretic Functions.'' §24.3.1 in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 826-827, 1972.

Hardy, G. H. and Wright, E. M. An Introduction to the Theory of Numbers, 5th ed. Oxford, England: Clarendon Press, pp. 257-258, 1979.