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mathematical analysis > calculus > series > Lambert series

number > number theory > analytic number theory > Lambert series

mathematical analysis > combinatorics > q-analog > Lambert series

algebra > combinatorics > q-analog > Lambert series

number > number theory > arithmetic function > Lambert series

Preferred term

Lambert series

Definition(s)

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

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

It can be resumed formally by expanding the denominator:

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

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

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

This series may be inverted by means of the Möbius inversion formula, and is an example of a Möbius transform.
(Wikipedia, The Free Encyclopedia, https://en.wikipedia.org/wiki/Lambert_series)

Broader concept(s)

Related concept(s)

Dirichlet character

In other languages

série de Lambert

French

URI

http://data.loterre.fr/ark:/67375/PSR-S5KNMR6X-4

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