Selberg's Formula

Author

Eric W. Weisstein
November 15, 2005

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

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

©2005 Wolfram Research, Inc. except for portions noted otherwise

Code

λ[d_,x_]:=MoebiusMu[d]Log[(x/d)]^2
f[n_,x_]:=f[n,x]=Plus@@(λ[#,x]&/@Divisors[n])
SelbergS[x_]:=Plus@@Table[f[n,x],{n,Floor[x]}]
SetAttributes[SelbergS,Listable]

Plot

<<Utilities`Typesetting`

Plot[{SelbergS[x],2x Log[x]},{x,0,100},PlotStyle->List/@{Red,Blue},AxesLabel->TraditionalForm/@{x,{StyleForm[S[x],FontColor->Red],StyleForm[2x Log[x],FontColor->Blue]}}]

Values

SelbergS[Range[5]]//FullSimplify

SelbergS[Range[10]]//Round