Hadamard Product

Author

Eric W. Weisstein
September 14, 2008

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

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

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

Analytic

((exp((ln(2 π)-(ℽ/2)-1) s))/(2 (s-1) Γ((s/2)+1)))(∏_n^∞)(((e^((4 s)/(4 ((Im((Subscript[ρ, n])))^2)+1))) (4 ((Im((Subscript[ρ, n])))^2)+(((1-2 s))^2)))/(4 ((Im((Subscript[ρ, n])))^2)+1))==ζ(s)

((Exp[(Log[2π]-1-EulerGamma/2)s])/(2(s-1)Gamma[1+s/2]))Product[(((E^((4 s)/(1+4 ((Im[ZetaZero[n]])^2)))) ((((1-2 s))^2)+4 ((Im[ZetaZero[n]])^2)))/(1+4 ((Im[ZetaZero[n]])^2))),{n,∞}]//Timing

Table[{Zeta[s],((Exp[(Log[2π]-1-EulerGamma/2)s])/(2(s-1)Gamma[1+s/2]))Product[(((E^((4 s)/(1+4 ((Im[ZetaZero[n]])^2)))) ((((1-2 s))^2)+4 ((Im[ZetaZero[n]])^2)))/(1+4 ((Im[ZetaZero[n]])^2))),{n,100}]}/.s->RandomComplex[],{1}]

((exp((ln(2 π)-(ℽ/2)-1) s) (∏_n^∞)(((e^((2 s)/(2 (Subscript[ρ, n]^2)-2 (Subscript[ρ, n])+1))) ((((1-2 (Subscript[ρ, n])))^2)+(((1-2 s))^2)))/((((1-2 (Subscript[ρ, n])))^2)+1)))/(2 (s-1) Γ((s/2)+1)))==ζ(s)

((Exp[(Log[2π]-1-EulerGamma/2)s])/(2(s-1)Gamma[1+s/2]))Product[(((E^((2 s)/(1-2 ZetaZero[n]+2 ((ZetaZero[n])^2)))) ((((1-2 s))^2)+(((1-2 ZetaZero[n]))^2)))/(1+(((1-2 ZetaZero[n]))^2))),{n,∞}]//Timing

(((E^((4 s)/(1+4 ((Im[ZetaZero[n]])^2)))) ((((1-2 s))^2)+4 ((Im[ZetaZero[n]])^2)))/(1+4 ((Im[ZetaZero[n]])^2)))/.Im[ZetaZero[n]]:>ZetaZero[n]-1/2//Simplify

Table[{Zeta[s],((Exp[(Log[2π]-1-EulerGamma/2)s])/(2(s-1)Gamma[1+s/2]))Product[(((E^((2 s)/(1-2 ZetaZero[n]+2 ((ZetaZero[n])^2)))) ((((1-2 s))^2)+(((1-2 ZetaZero[n]))^2)))/(1+(((1-2 ZetaZero[n]))^2))),{n,100}]}/.s->RandomComplex[],{1}]

((exp((ln(2 π)-(ℽ/2)-1) s))/(2 (s-1) Γ((s/2)+1)))(Underoverscript[∏', (n=-∞), ∞])(1-(s/(Subscript[ρ, n]))) exp((s/(Subscript[ρ, n])))==ζ(s)

TimeConstrained[((Exp[(Log[2π]-1-EulerGamma/2)s])/(2(s-1)Gamma[1+s/2]))Product[Piecewise[{{(1-(s/(ZetaZero[n])))Exp[s/ZetaZero[n]],n≠0}},1],{n,-∞,∞}],60]//Timing

Table[{Zeta[s],((Exp[(Log[2π]-1-EulerGamma/2)s])/(2(s-1)Gamma[1+s/2]))Product[Piecewise[{{(1-(s/(ZetaZero[n])))Exp[s/ZetaZero[n]],n≠0}},1],{n,-50,50}]}/.s->RandomComplex[],{1}]

TimeConstrained[Product[Piecewise[{{(1-(s/(ZetaZero[n])))Exp[s/ZetaZero[n]],n≠0}}],{n,-∞,∞}],60]//Timing

Plot

Length[zeros=ReadList["/Volumes/Users/Data/Math/ZetaZeros.txt"]]//Timing

zeros=Im[ZetaZero[Range[10^2]]//N];//Timing

zeros=Im[ZetaZero[Range[10^3]]//N];//Timing

zeros=Im[ZetaZero[Range[10^4]]//N];//Timing

zeros=Im[ZetaZero[Range[10^5]]//N];//Timing

Zeros and their complex conjugates

FullSimplify[((1-(s/(1/2+I x)))Exp[s/(1/2+I x)](1-(s/(1/2-I x)))Exp[s/(1/2-I x)])]

zeta[s_,n_]:=((Exp[(Log[2π]-1-EulerGamma/2)s])/(2(s-1)Gamma[1+s/2]))Times@@(((((E^((4 s)/(1+4 ((#)^2)))) ((((1-2 s))^2)+4 ((#)^2)))/(1+4 ((#)^2))))&/@Take[zeros,n])

V5

Plot[Evaluate[Zeta[s]-zeta[s,#]&/@{100,500,1000,2000}],{s,-1,2},PlotStyle->{Red,Yellow,Green,Blue},AxesLabel->TraditionalForm/@{x,Zeta[x]-Subscript[ζ,n][x]}]

V7

Plot[Evaluate[Zeta[s]-zeta[s,#]&/@{100,500,1000,2000}],{s,-1,2},PlotStyle->{Red,Yellow,Green,Blue},AxesLabel->{x,Zeta[x]-Subscript[ζ,n][x]},Exclusions->1]

...where'd the red curve go?...

Constants

N[(Log[2π]-1-EulerGamma/2),10]

N[(Log[4π]/2-1-EulerGamma/2),10]