Nint Zeta Function

Author

Eric W. Weisstein
November 29, 2008

This notebook downloaded from http://mathworld.wolfram.com/notebooks/SpecialFunctions/NintZetaFunction.nb.

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

©2008 Wolfram Research, Inc. except for portions noted otherwise

Code

NintZetaS[N_,s_]:=Sum[((Round[(n^(1/N))])^(-s)),{n,∞}]
NNintZetaS[NN_,s_]:=Total[Table[((Round[(n^(1./NN))])^(-s)),{n,10^6}]//N]

Formulas

NintZetaS[N,s]//Timing

S[2,s]

NintZetaS[2,s]

With[{s=RandomReal[{3,10}]},{2Zeta[s-1],NNintZetaS[2,s]}]

S[3,s]

NintZetaS[3,s]

With[{s=RandomReal[{3,10}]},{3Zeta[s-2]+(4^(-s))Zeta[s],NNintZetaS[3,s]}]

S[4,s]

NintZetaS[4,s]

With[{s=RandomReal[{3,10}]},{4Zeta[s-3]+Zeta[s-1],NNintZetaS[4,s]}]

Integers

NintZetaS[3,4]

ListPlot[With[{NN=3,s=4},Table[((Round[(n^(1/NN))])^(-s)),{n,10^4}]]//N]

{NNintZetaS[3,4],((π^2)/2)+((π^4)/23046)}//N

NintZetaS[5,6]

{NNintZetaS[5,6],((5(π^2))/6)+((π^4)/36)+((π^6)/(4^12))((1/945)-((170912+49928(Sqrt[2]))/25)(Sqrt[1-(Sqrt[(1/2)])]))//N}

NintZetaS[6,7]

{NNintZetaS[6,7],(π^2)+((π^4)/18)+((π^6)/2520)+((246013+353664(Sqrt[2]))/45)((π^7)/(2^27))//N}