Belt (mechanical)

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

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

It can be resummed formally by expanding the denominator:

$S(q)=\sum _{n=1}^{\mathrm{\infty }}{a}_n\sum _{k=1}^{\mathrm{\infty }}{q}^{nk}=\sum _{m=1}^{\mathrm{\infty }}{b}_m{q}^m$

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

$\sum _{n=1}^{\mathrm{\infty }}{q}^n{\sigma }_0(n)=\sum _{n=1}^{\mathrm{\infty }}\frac{q^n}{1-q^n}$

where ${\sigma }_0(n)=d(n)$ is the number of positive divisors of the number n.

For the higher order sigma functions, one has

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

where $\alpha$ is any complex number and

${\sigma }_{\alpha }(n)=({\text{<mrow data-mjx-texclass="ORD">Id</mrow>}}_{\alpha }*1)(n)=\sum _{d\mid n}{d}^{\alpha }$

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 $\mu (n)$:

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

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

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

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

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

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

$\sum _{n=1}^{\mathrm{\infty }}\frac{a_n}{e^{zn}-1}=\sum _{m=1}^{\mathrm{\infty }}{b}_m{e}^{-mz}$

where

$b_m=(a*1)(m)=\sum _{n\mid m}{a}_n$

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 $q^n/(1-q^n)={\mathrm{L}\mathrm{i}}_0(q^n)$ is a polylogarithm function, we may refer to any sum of the form

$\sum _{n=1}^{\mathrm{\infty }}\frac{{\xi }^n{\mathrm{L}\mathrm{i}}_u(\alpha q^n)}{n^s}=\sum _{n=1}^{\mathrm{\infty }}\frac{{\alpha }^n{\mathrm{L}\mathrm{i}}_s(\xi q^n)}{n^u}$

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

$12{(\sum _{n=1}^{\mathrm{\infty }}{n}^2{\mathrm{L}\mathrm{i}}_{-1}(q^n))}^2=\sum _{n=1}^{\mathrm{\infty }}{n}^2{\mathrm{L}\mathrm{i}}_{-5}(q^n)-\sum _{n=1}^{\mathrm{\infty }}{n}^4{\mathrm{L}\mathrm{i}}_{-3}(q^n),$

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

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