Lambert series

The series

n=1anzn1-zn=a1z1-z+a2z21-z2+ (1)

is called Lambert seriesMathworldPlanetmath.  We here consider more closely only the special case

n=1xn1-xn=x1-x+x21-x2+ (2)

for the real x.

I.  Convergence

1.x=±1:  The series is not defined.

2.|x|>1:  We have


whence the series (2) diverges.

3.0x<1:  The series with nonnegative terms converges, since


4.-1<x<0:  We get an alternating seriesMathworldPlanetmath with

|xn1-xn|=|x|n|1-xn||x|n1-|x|n|x|n1-|x| 0asn,

and by Leibniz theorem, the series converges.

Thus we have the result that the Lambert series (2) converges, absolutely, when  |x|<1.

Let  |x|<1.  the terms to geometric seriesMathworldPlanetmath:
x1-x = x + x2 + x3 + x4 + x5 + x6 + x21-x2 = x2 + x4 + x6 + x31-x3 = x3 + x6 + x41-x4 = x4 + x51-x5 = x5 + x61-x6 = x6 +

Those geometric series converge absolutely,


and the series k=1|x|k1-|x|k converges.  Thus we can sum the geometric series by the columns:


Apparently, the coefficient of any xk in this power seriesMathworldPlanetmath expresses, by how many positive integers the number k is divisible, i.e. the coefficient is given by the divisor functionMathworldPlanetmath τ.  So we may write the power series form of the Lambert series as

Title Lambert series
Canonical name LambertSeries
Date of creation 2013-03-22 18:46:42
Last modified on 2013-03-22 18:46:42
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 13
Author pahio (2872)
Entry type Example
Classification msc 30B10
Classification msc 40A05
Related topic NecessaryConditionOfConvergence
Related topic CauchysRootTest
Related topic TauFunction