Euler Product

Author

Eric W. Weisstein
August 8, 2008

This notebook downloaded from http://mathworld.wolfram.com/notebooks/PrimeNumbers/EulerProduct.nb.

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

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

Identities

ζ(s)==(∏_n^∞)(1/(1-(1/(Subscript[p, n]^s))))

V5

Product[(((1-(1/((Prime[n])^s))))^(-1)),{n,∞}]

V7

Product[1/(1-1/Prime[k]^s),{k,∞}]

Zeta/@Range[2,5]//N

primes=Table[Prime[k],{k,500}];

Table[Fold[#1/(1-1/#2^s)&,1,primes],{s,2,5}]//N

(∑_n^∞)((μ(n))/(n^s))==(1/(ζ(s)))

Sum[((MoebiusMu[n])/(n^s)),{n,∞}]

Product Construction

Rest[FoldList[#1/(1-1/#2)&,1,Table[Subscript[p,k],{k,4}]]]//TraditionalForm

prods[n_]:=FoldList[#1/(1-1/#2)&,1,Table[Prime[k],{k,n}]]

prods[10]

A060753/A038110

Table[ Product[ 1-1/Prime[ k ],{k,n-1} ]/Prime[ n],{n,1,10}]

eg=MapIndexed[#1/Log@@Prime[#2]&,Rest[prods[200]]];

%//N

Exp[EulerGamma]//N

<<Utilities`Plot`

With[{max=2000,eg=Exp[EulerGamma]},
IntegerListPlot[MapIndexed[#1/Log@@Prime[#2]&,Rest[prods[max]]],PlotStyle->Red,Center->True,
AxesLabel->TraditionalForm/@{n,(1/(ln (Subscript[p, n])))(∏_(i=1)^n)(1/(1-(1/(Subscript[p, i]))))},
Epilog->{Blue,Text[TraditionalForm[eg],{100,1.7815}],Line[{{0,#},{1.1max,#}}]&[eg]}]
]//Timing

1/Zeta[s]

Expand[(1-(1/(Subscript[p, 1])))(1-(1/(Subscript[p, 2])))(1-(1/(Subscript[p, 3])))(1-(1/(Subscript[p, 4])))]

<<DiscreteMath`Combinatorica`

1+Plus@@(1/Times@@#(-1)^Length[#]&/@Map[Subscript[p,#]&,Flatten[KSubsets[Range[4],#]&/@Range[4],1],{2}])