Prime Products

Author

Eric W. Weisstein
August 31, 2008

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

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

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

Mertens Theorem

(lim_(n->∞))(1/(ln((Subscript[p, n])))) (∏_(k=1)^n)(1/(1-(1/(Subscript[p, k]))))==(e^γ)

V6

Limit[(1/(Log[Prime[n]]))Product[(1/(1-(1/(Prime[k])))),{k,n}],n->∞]

N[Times@@Table[1/(1-1/Prime[k]),{k,#}]/Log[Prime[#]]&[10000],20]//Timing

N[Times@@Table[1/(1-1/Prime[k]),{k,#}]/Log[Prime[#]]&[50000],20]//Timing

N[Exp[EulerGamma],20]

V7

Limit[(1/(Log[Prime[n]]))Product[(1/(1-(1/(Prime[k])))),{k,n}],n->∞]

Mertens Theorem Extension

N[Times@@Table[1/(1+1/Prime[k]),{k,#}]Log[Prime[#]]&[10000],20]//Timing

N[Times@@Table[1/(1+1/Prime[k]),{k,#}]Log[Prime[#]]&[100000],20]//Timing

N[Pi^2/(6Exp[EulerGamma]),20]

Products

(∏_k^n)(Subscript[p, k])== (Subscript[p, n])#

Product[Prime[k],{k,n}]

(∏_k^∞)(((Subscript[p, k]^2)+1)/((Subscript[p, k]^2)-1))==(5/2)

Ramanujan's Product

V6

Product[(Prime[k]^2+1)/(Prime[k]^2-1),{k,∞}]

N[Times@@Table[(Prime[k]^2+1)/(Prime[k]^2-1),{k,10000}],20]//Timing

N[Times@@Table[(Prime[k]^2+1)/(Prime[k]^2-1),{k,50000}],20]//Timing

V7

Product[(Prime[k]^2+1)/(Prime[k]^2-1),{k,∞}]

(∏_n^∞)((1/(Subscript[p, n]^s))+1)==((ζ(s))/(ζ(2 s)))

V6

Product[1+1/Prime[n]^s,{n,∞}]

Product[1+1/Prime[n]^#,{n,∞}]&/@Range[2,10,2]

V7

Product[1+1/Prime[n]^s,{n,∞}]

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

V6

Product[1-1/Prime[n]^s,{n,∞}]

Product[1-1/Prime[n]^#,{n,∞}]&/@Range[2,10,2]

V7

Product[1-1/Prime[n]^s,{n,∞}]

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

V7

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

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

(∏_(n=2)^∞)(((χ(n))/(Subscript[p, n]))+1)==(2/π)

Oakes

V6

Chi[p_]:=2-Mod[p,4]

Times@@(1+Chi[#]/#&/@Prime/@Range[10^5])//N

N[2/Pi,10]

Times@@(1/(1-Chi[#]/#)&/@Prime/@Range[10^5])//N//Timing

N[Pi/4,10]

V7

Product[1+((2-Mod[Prime[n],4])/(Prime[n])),{n,2,∞}]

(∏_(n=2)^∞)(1-((χ(n))/(Subscript[p, n])))==(4/π)

V7

Product[1-((2-Mod[Prime[n],4])/(Prime[n])),{n,2,∞}]

(∏_k^∞)(1-(2/(Subscript[p, k]^2)))==0.32263459147223017`...

(Subscript[Q, 2])(n)

V6

Product[1-(2/((Prime[k])^2)),{k,∞}]

Product[1-(2/((Prime[k])^2)),{k,10^4}]//N

V7

Product[1-(2/((Prime[k])^2)),{k,∞}]

Product[1-(2/((Prime[k])^2)),{k,10^4}]//N

NProduct[1-(2/((Prime[k])^2)),{k,∞}]

(∏_k^∞)(1/(((sin(((π (Subscript[p, k]))/2)))/(Subscript[p, k]))+1))==(π/2)

TimeConstrained[Product[(((1+((Sin[(π/2)Prime[k]])/(Prime[k]))))^(-1)),{k,∞}],60]//InputForm

Product[(((1+((Sin[(π/2)Prime[k]])/(Prime[k]))))^(-1)),{k,10^4}]//N

π/2.

(∏_(n=2)^∞)(1/(((((-1))^((1/2) ((Subscript[p, n])-1)))/(Subscript[p, n]))+1))==(π/2)

Product[(((1+((((-1))^((Prime[n]-1)/2))/(Prime[n]))))^(-1)),{n,2,∞}]//InputForm

Product[(((1+((((-1))^((Prime[n]-1)/2))/(Prime[n]))))^(-1)),{n,2,10^4}]//N

Regularized Products

(SuperStar[(∏_k^∞)])(Subscript[p, k])==4 (π^2)

Product[Prime[k],{k,∞}]

Product[Prime[k],{k,∞},Regularization->"Dirichlet"]

(SuperStar[(∏_k^∞)])((Subscript[p, k])^s)==(((2π))^(2s))

V6

Product[((Prime[k])^s),{k,∞}]

V7

Product[((Prime[k])^s),{k,∞}]

Product[((Prime[k])^s),{k,∞},Regularization->"Dirichlet"]

(SuperStar[(∏_n^∞)])((Subscript[p, n]^s)-1)==((((2 π))^(2 s))/(ζ(s)))

Product[((Prime[n])^s)-1,{n,∞}]

Product[((Prime[n])^s)-1,{n,∞},Regularization->"Dirichlet"]//InputForm

Table[((((2π))^(2s))/(Zeta[s])),{s,5}]