- Lambert series
In

mathematics , a**Lambert series**, named forJohann 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 aMöbius 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}aretrigonometric function s, for example, "a"_{n}=sin (2"n" "x"), can be evaluated by various combinations of thelogarithmic derivative s of Jacobitheta function s.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; seeZeta 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 byBruce Berndt .**ee also***

Erdős–Borwein constant

*Appell-Lerch sum (sometimes called generalized Lambert series).**References***

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**Lambert summation**— In mathematical analysis, Lambert summation is a summability method for a class of divergent series. DefinitionA series sum a n is Lambert summable to A , written sum a n = A (mathrm{L}), if:lim {r ightarrow 1 } (1 r) sum {n=1}^infty frac{n a n… … Wikipedia**Lambert Le Bègue**— • Priest and reformer, lived at Liège, Belgium, about the middle of the twelfth century Catholic Encyclopedia. Kevin Knight. 2006. Lambert Le Begue Lambert Le Bègue … Catholic encyclopedia**Lambert of Hersfeld**— • A medieval historian; b. in Franconia or Thuringia, c. 1024; d. after 1077 Catholic Encyclopedia. Kevin Knight. 2006. Lambert of Hersfeld Lambert of Hersfeld … Catholic encyclopedia**Lambert Bartak**— is the full time organist for the NCAA Division I College World Series at Johnny Rosenblatt Stadium in Omaha, Nebraska. Lambert has played full time for the series since 1988; he first played the organ for the event in the 1950s.Fact|date=March… … Wikipedia**Lambert Gardens**— was a private botanical garden of over 30 acres (120,000 m sup2;) in the Reed neighborhood of Portland, Oregon, north of Reed College at SE 28th Ave. and SE Steele St. It was a significant attraction, drawing tens of thousands of visitors a… … Wikipedia**Lambert W function**— The graph of W(x) for W > −4 and x < 6. The upper branch with W ≥ −1 is the function W0 (principal branch), the lower branch with W ≤ −1 is the function W−1. In mathematics, the Lambert W function, also called the Omega function or product… … Wikipedia**Lambert**— Cette page d’homonymie répertorie les différents sujets et articles partageant un même nom. Pour les articles homonymes, voir Saint Lambert. Lambert est un nom propre qui peut désigner : Sommaire … Wikipédia en Français**Lambert, Hendricks, & Ross!**— Infobox Album Name = Lambert, Hendricks, Ross! Type = Studio album Artist = Lambert, Hendricks Ross Released = 1960 Recorded = Genre = Length = 122:58 Label = Columbia Producer = Reviews = * Allmusic Rating|5|5… … Wikipedia**LAMBERT, George Washington Thomas (1873-1930), the third name was never used**— artist was born at St Petersburg, Russia, on 13 September 1873, the fourth child and only son of George Washington Lambert, an American engineer who went to Russia to assist in the construction of railways. His mother was the only child of an… … Dictionary of Australian Biography**Lambert Wilson**— Infobox actor name = Lambert Wilson caption = Lambert Wilson, January 2008 birthname = birthdate = birth date and age|1958|8|3 birthplace = Neuilly sur Seine, France othername = occupation = yearsactive = 1982 present spouse = domesticpartner =… … Wikipedia