|
|
| (One intermediate revision by one other user not shown) |
| Line 1: |
Line 1: |
| {{For|generalized Lambert series|Appell–Lerch sum}}
| | Hi there. Let me begin by introducing the writer, her title is Myrtle Cleary. Bookkeeping is her working day occupation now. To collect badges is what her family and her appreciate. California is our beginning location.<br><br>Here is my web site ... [http://peopleinput.biz/archafrique/understand-more-about-yeast-infections-and-preventing-them home std test kit] |
| | |
| In [[mathematics]], a '''Lambert series''', named for [[Johann Heinrich Lambert]], is a [[Series (mathematics)|series]] taking the form
| |
| | |
| :<math>S(q)=\sum_{n=1}^\infty a_n \frac {q^n}{1-q^n}.</math>
| |
| | |
| It can be resummed [[Formal series|formally]] by expanding the denominator:
| |
| | |
| :<math>S(q)=\sum_{n=1}^\infty a_n \sum_{k=1}^\infty q^{nk} = \sum_{m=1}^\infty b_m q^m </math>
| |
| | |
| where the coefficients of the new series are given by the [[Dirichlet convolution]] of ''a''<sub>''n''</sub> with the constant function 1(''n'') = 1:
| |
| | |
| :<math>b_m = (a*1)(m) = \sum_{n\mid m} a_n. \,</math>
| |
| | |
| 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
| |
| | |
| :<math>\sum_{n=1}^\infty q^n \sigma_0(n) = \sum_{n=1}^\infty \frac{q^n}{1-q^n}</math>
| |
| | |
| where <math>\sigma_0(n)=d(n)</math> is the number of positive [[divisor function|divisors]] of the number ''n''.
| |
| | |
| For the higher order [[divisor function|sigma functions]], one has
| |
| | |
| :<math>\sum_{n=1}^\infty q^n \sigma_\alpha(n) = \sum_{n=1}^\infty \frac{n^\alpha q^n}{1-q^n}</math>
| |
| | |
| where <math>\alpha</math> is any [[complex number]] and
| |
| | |
| :<math>\sigma_\alpha(n) = (\textrm{Id}_\alpha*1)(n) = \sum_{d\mid n} d^\alpha \,</math>
| |
| is the divisor function. | |
| | |
| Lambert series in which the ''a''<sub>''n''</sub> are [[trigonometric function]]s, for example, ''a''<sub>''n''</sub> = sin(2''n'' ''x''), can be evaluated by various combinations of the [[logarithmic derivative]]s of Jacobi [[theta function]]s.
| |
| | |
| Other Lambert series include those for the [[Möbius function]] <math>\mu(n)</math>:
| |
| | |
| :<math>\sum_{n=1}^\infty \mu(n)\,\frac{q^n}{1-q^n} = q.</math>
| |
| | |
| For [[Euler's totient function]] <math>\phi(n)</math>:
| |
| :<math>\sum_{n=1}^\infty \varphi(n)\,\frac{q^n}{1-q^n} = \frac{q}{(1-q)^2}.</math>
| |
| | |
| For [[Liouville's function]] <math>\lambda(n)</math>:
| |
| | |
| :<math>\sum_{n=1}^\infty \lambda(n)\,\frac{q^n}{1-q^n} =
| |
| \sum_{n=1}^\infty q^{n^2}</math>
| |
| | |
| with the sum on the left similar to the [[Ramanujan theta function]].
| |
| | |
| ==Alternate form==
| |
| Substituting <math>q=e^{-z}</math> one obtains another common form for the series, as
| |
| | |
| :<math>\sum_{n=1}^\infty \frac {a_n}{e^{zn}-1}= \sum_{m=1}^\infty b_m e^{-mz}</math> | |
| | |
| where
| |
| :<math>b_m = (a*1)(m) = \sum_{n\mid m} a_n\,</math>
| |
| | |
| as before. Examples of Lambert series in this form, with <math>z=2\pi</math>, 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 <math>q^n/(1 - q^n ) = \mathrm{Li}_0(q^{n})</math> is a [[polylogarithm]] function, we may refer to any sum of the form
| |
| | |
| :<math>\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}</math>
| |
| | |
| as a Lambert series, assuming that the parameters are suitably restricted. Thus
| |
| | |
| :<math>12\left(\sum_{n=1}^{\infty} n^2 \, \mathrm{Li}_{-1}(q^n)\right)^{\!2} = \sum_{n=1}^{\infty}
| |
| n^2 \,\mathrm{Li}_{-5}(q^n) -
| |
| \sum_{n=1}^{\infty} n^4 \, \mathrm{Li}_{-3}(q^n),</math>
| |
| | |
| 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==
| |
| * {{cite book|last=Berry|first=Michael V. |title=Functions of Number Theory|year=2010|publisher=CAMBRIDGE UNIVERSITY PRESS|isbn=978-0-521-19225-5|pages=637–641|url=http://dlmf.nist.gov}}
| |
| * {{cite journal|first1=Preston A. | last1=Lambert
| |
| |title=Expansions of algebraic functions at singular points
| |
| |journal=Proc. Am. Philos. Soc.
| |
| |year=1904
| |
| |volume=43 | number=176
| |
| |jstor=983503
| |
| |pages=164–172
| |
| }}
| |
| * {{Apostol IANT}}
| |
| * {{springer|title=Lambert series|id=p/l057340}}
| |
| * {{mathworld|urlname=LambertSeries|title=Lambert Series}}
| |
| | |
| [[Category:Analytic number theory]]
| |
| [[Category:Q-analogs]]
| |
| [[Category:Mathematical series]]
| |
Hi there. Let me begin by introducing the writer, her title is Myrtle Cleary. Bookkeeping is her working day occupation now. To collect badges is what her family and her appreciate. California is our beginning location.
Here is my web site ... home std test kit