I was recently intrigued by the fact that Sum[MoebiusMu[n]/n^s,{n,1,Infinity}] = 1/Zeta[s]. Since MoebiusMu[n] = -1, 0, or 1 ... and since there is a 1/Zeta[2] chance that it will be -1 or 1, I decided to look at how the numbers split up between -1 and 1. Define (for k from 1 to infinity), where Mu is the MoebiusMu function. My actual Mathematica code is at the bottom of the message. MuRatio[s] = Sum[If[Mu[k]=1,1,0]/k^s]/Sum[If[Mu[k]=-1,1,0]/k^s] MuRatio[2] = Sum[1/{k:Mu[k]==1}^2]/Sum[1/{k:Mu[k]==-1}^2] Then MuRatio[2] = 7/3 MuRatio[4] = 13 MuRatio[6] = 703/12 MuRatio[8] = 14527/59 MuRatio[10] = 524354/521 MuRatio[12] = 354333857/872492 MuRatio[14] = 6785975897/415603 Also Sum[Abs[Mu[k]]/k^2] = 15/Pi^2 Sum[1/{k:Mu[k]==1}^2] = 21/(2 Pi^2) Sum[1/{k:Mu[k]==1}^2] = 9/(2 Pi^2) --Ed Pegg Jr, www.mathpuzzle.com MuRatio[s_] := Total[Table[N[If[MoebiusMu[k]== 1,1,0]/k^s, 50], {k, 1, 1000000}]]/ Total[Table[N[If[MoebiusMu[k]==-1,1,0]/k^s, 50], {k, 1, 1000000}]]