Lambert series

For generalized Lambert series, see Appell–Lerch sum.

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

$S(q)=\sum _{n=1}^{\infty }a_{n}{\frac {q^{n}}{1-q^{n}}}.$

It can be resummed formally by expanding the denominator:

$S(q)=\sum _{n=1}^{\infty }a_{n}\sum _{k=1}^{\infty }q^{nk}=\sum _{m=1}^{\infty }b_{m}q^{m}$

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

$b_{m}=(a*1)(m)=\sum _{n\mid m}a_{n}.\,$

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

Examples

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

$\sum _{n=1}^{\infty }q^{n}\sigma _{0}(n)=\sum _{n=1}^{\infty }{\frac {q^{n}}{1-q^{n}}}$

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

For the higher order sigma functions, one has

$\sum _{n=1}^{\infty }q^{n}\sigma _{\alpha }(n)=\sum _{n=1}^{\infty }{\frac {n^{\alpha }q^{n}}{1-q^{n}}}$

where $\alpha$ is any complex number and

$\sigma _{\alpha }(n)=({\textrm {Id}}_{\alpha }*1)(n)=\sum _{d\mid n}d^{\alpha }\,$

is the divisor function.

Lambert series in which the a_n are trigonometric functions, for example, a_n = sin(2n x), can be evaluated by various combinations of the logarithmic derivatives of Jacobi theta functions.

Other Lambert series include those for the Möbius function $\mu (n)$:

$\sum _{n=1}^{\infty }\mu (n)\,{\frac {q^{n}}{1-q^{n}}}=q.$

For Euler's totient function $\phi (n)$:

$\sum _{n=1}^{\infty }\varphi (n)\,{\frac {q^{n}}{1-q^{n}}}={\frac {q}{(1-q)^{2}}}.$

For Liouville's function $\lambda (n)$:

$\sum _{n=1}^{\infty }\lambda (n)\,{\frac {q^{n}}{1-q^{n}}}=\sum _{n=1}^{\infty }q^{n^{2}}$

with the sum on the left similar to the Ramanujan theta function.

Alternate form

Substituting $q=e^{-z}$ one obtains another common form for the series, as

$\sum _{n=1}^{\infty }{\frac {a_{n}}{e^{zn}-1}}=\sum _{m=1}^{\infty }b_{m}e^{-mz}$

where

$b_{m}=(a*1)(m)=\sum _{d\mid m}a_{d}\,$

as before. Examples of Lambert series in this form, with $z=2\pi$, occur in expressions for the Riemann zeta function for odd integer values; see Zeta constants for details.

Current usage

In the literature we find Lambert series applied to a wide variety of sums. For example, since $q^{n}/(1-q^{n})=\mathrm {Li} _{0}(q^{n})$ is a polylogarithm function, we may refer to any sum of the form

$\sum _{n=1}^{\infty }{\frac {\xi ^{n}\,\mathrm {Li} _{u}(\alpha q^{n})}{n^{s}}}=\sum _{n=1}^{\infty }{\frac {\alpha ^{n}\,\mathrm {Li} _{s}(\xi q^{n})}{n^{u}}}$

as a Lambert series, assuming that the parameters are suitably restricted. Thus

$12\left(\sum _{n=1}^{\infty }n^{2}\,\mathrm {Li} _{-1}(q^{n})\right)^{\!2}=\sum _{n=1}^{\infty }n^{2}\,\mathrm {Li} _{-5}(q^{n})-\sum _{n=1}^{\infty }n^{4}\,\mathrm {Li} _{-3}(q^{n}),$

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.

See also

• Erdős–Borwein constant

References

• 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
• "Lambert series", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
• Weisstein, Eric W. "Lambert Series". MathWorld.