Lambert Series

Author

Eric W. Weisstein
March 15, 2011

This notebook downloaded from http://mathworld.wolfram.com/notebooks/NumberTheoreticFunctions/LambertSeries.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/LambertSeries.html.

©2011 Wolfram Research, Inc. except for portions noted otherwise

Examples

(∑_(n=1)^∞)((μ(n) (x^n))/(1-(x^n)))==x

v5.2

Sum[((MoebiusMu[n](x^n))/(1-(x^n))),{n,∞}]

v6

Sum[((MoebiusMu[n](x^n))/(1-(x^n))),{n,∞}]

v7

Sum[((MoebiusMu[n](x^n))/(1-(x^n))),{n,∞}]

(∑_(n=1)^∞)((ϕ(n) (x^n))/(1-(x^n)))==(x/(((1-x))^2))

v5.2

Sum[((EulerPhi[n](x^n))/(1-(x^n))),{n,∞}]

v6

Sum[((EulerPhi[n](x^n))/(1-(x^n))),{n,∞}]

v7

Sum[((EulerPhi[n](x^n))/(1-(x^n))),{n,∞}]

(∑_(n=1)^∞)((x^n)/(1-(x^n)))==(∑_(n=1)^∞)(DivisorSigma[0, n]) (x^n)==(((QPolyGamma[1, x])+log(1-x))/(log(x)))

V5.2

Sum[((x^n)/(1-(x^n))),{n,∞}]

V6

Sum[((x^n)/(1-(x^n))),{n,∞}]

V7

Sum[((x^n)/(1-(x^n))),{n,∞}]

V8

Sum[((x^n)/(1-(x^n))),{n,∞}]

f[x_?NumericQ]:=NSum[((x^n)/(1-(x^n))),{n,∞}]

Plot[f[x]-((QPolyGamma[1,x]+Log[1-x])/(Log[x])),{x,.1,.9}]

(∑_(n=1)^∞)(((n^k)(x^n))/(1-(x^n)))==(∑_(n=1)^∞)(Subscript[σ, k])(n) (x^n)

(∑_(n=0)^∞)(( (((-1))^n)(x^(2 n+1)))/(1-(x^(2 n+1))))==(1/4) ((((Subscript[ϑ, 3])(0,x))^2)-1)

v5.2

Sum[((x^(2n+1))/(1-(x^(2n+1))))(((-1))^n),{n,0,∞}]

v6

Sum[((x^(2n+1))/(1-(x^(2n+1))))(((-1))^n),{n,0,∞}]

v7

Sum[((x^(2n+1))/(1-(x^(2n+1))))(((-1))^n),{n,0,∞}]

(∑_(n=1)^∞)((λ(n) (x^n))/(1-(x^n)))==(1/2) ((Subscript[ϑ, 3])(0,x)-1)

Sum[(x^(n^2)),{n,∞}]//Simplify

Series[(1/2) (-1+EllipticTheta[3,0,x]),{x,0,50}]

Series[Sum[((LiouvilleLambda[n](x^n))/(1-(x^n))),{n,100}],{x,0,50}]

Position[CoefficientList[%,x],1]-1//Flatten

V7

Sum[((LiouvilleLambda[n](x^n))/(1-(x^n))),{n,∞}]

Borwein and Borwein, p. 91

L

L[b_]:=Sum[((b^n)/(1-(b^n))),{n,∞}]

L[b]

(∑_n^∞)(1/((β^(-n))-1))==(((QPolyGamma[1, β])+log(1-β))/(log(β)))

Sum[(1/((β^(-n))-1)),{n,∞}]

With[{β=.1},Sum[(1/((β^(-n))-1)),{n,1000}]]

With[{x=.1},((QPolyGamma[1,x]+Log[1-x])/(Log[x]))]

(∑_n^∞)(1/((β^(-n))+(β^n)))==(1/4) ((((Subscript[ϑ, 3])(0,β))^2)-1)

Sum[(1/((β^n)+(β^(-n)))),{n,∞}]

With[{β=.2},{NSum[(1/((β^n)+(β^(-n)))),{n,∞}],(-1+EllipticTheta[3,0,β]^2)/4}]

With[{β=.8},{NSum[(1/((β^n)+(β^(-n)))),{n,∞}],(-1+EllipticTheta[3,0,β]^2)/4}]