Lambert series

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 $1(n)=1$::$b\_m\; =\; (a*1)(m)\; =\; sum\_\{nmid\; m\}\; a\_n\; ,$

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 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:$sigma\_alpha(n)\; =\; (\; extrm\{Id\}\_alpha*1)(n)\; =\; sum\_\{dmid\; n\}\; d^alpha\; ,$is the divisor function.

Lambert series in which the "a"_n are trigonometric functions, for example, "a"_n=sin (2"n" "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:$b\_m\; =\; (a*1)(m)\; =\; sum\_\{nmid\; m\}\; a\_n,$

as before. Examples of Lambert series in this form, with $z=2pi$, 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

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

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.

ee also

* Erdős–Borwein constant
*Appell-Lerch sum (sometimes called generalized Lambert series).

References

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