## Lambert Series

A series of the form

 (1)

for . Then
 (2)

where
 (3)

Some beautiful series of this type include
 (4) (5) (6) (7) (8)

where is the Möbius Function, is the Totient Function, is the number of divisors of , is the Divisor Function, and is the number of representations of in the form where and 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.