It can be resummed formally by expanding the denominator:
where the coefficients of the new series are given by the Dirichlet convolution of an with the constant function 1(n) = 1:
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 is the number of positive divisors of the number n.
For the higher order sigma functions, one has
where is any complex number and
is the divisor function.
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.
Substituting one obtains another common form for the series, as
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 q 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.
- Berry, Michael V. (2010). Functions of Number Theory. CAMBRIDGE UNIVERSITY PRESS. pp. 637–641. ISBN 978-0-521-19225-5.
- Lambert, Preston A. (1904). "Expansions of algebraic functions at singular points". Proc. Am. Philos. Soc. 43 (176): 164–172. JSTOR 983503.
- Apostol, Tom M. (1976), Introduction to analytic number theory, Undergraduate Texts in Mathematics, New York-Heidelberg: Springer-Verlag, ISBN 978-0-387-90163-3, MR 0434929, Zbl 0335.10001
- Hazewinkel, Michiel, ed. (2001), "Lambert series", Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4
- Weisstein, Eric W., "Lambert Series", MathWorld.