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

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

It can be resummed formally by expanding the denominator:

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

where the coefficients of the new series are given by the Dirichlet convolution of $\{a\_n\}$ with the constant function $1(n)=1$:

$b\_m\; =\; (a*1)(m)\; =\; \backslash sum\_\{n|m\}\; a\_n$

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

Examples

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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

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

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

For the higher order sigma functions, one has

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

where $\backslash alpha$ is any complex number and

$\backslash sigma\_\backslash alpha(n)\; =\; (\backslash textrm\{Id\}\_\backslash alpha*1)(n)\; =\; \backslash sum\_\{d|n\}\; d^\backslash alpha$

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 Mobius function $\backslash mu(n)$:

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

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

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

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

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

\sum_{n=1}^\infty q^{n^2}

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

Alternate form

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

$\backslash sum\_\{n=1\}^\backslash infty\; \backslash frac\; \{a\_n\}\{e^\{zn\}-1\}=\; -\; \backslash sum\_\{m=1\}^\backslash infty\; b\_m\; e^\{-mz\}$

where

$b\_m\; =\; (a*1)(m)\; =\; \backslash sum\_\{n|m\}\; a\_n$

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

See also

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• Erdős–Borwein constant

References

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• Tom M. Apostol, Introduction to Analytic Number Theory, (1976) Springer-Verlag, New York. ISBN 0387901639

Analytic number theory | Q-analogs | Mathematical series

Série de Lambert