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A131671
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Decimal expansion of prime analog of the Kepler-Bouwkamp constant: Product_{k>=2} cos(Pi/prime(k)).
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3
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3, 1, 2, 8, 3, 2, 9, 2, 9, 5, 0, 8, 8, 8, 1, 8, 3, 8, 3, 3, 3, 2, 5, 9, 3, 6, 3, 9, 6, 8, 5, 3, 6, 4, 2, 1, 7, 5, 6, 8, 3, 3, 6, 8, 7, 7, 6, 7, 1, 1, 7, 3, 8, 5, 3, 1, 9, 8, 6, 5, 1, 3, 0, 1, 9, 7, 6, 7, 9, 7, 2, 6, 1, 9, 0, 7, 0, 3, 4, 8, 1, 3, 0, 7, 6, 2, 3, 3, 2, 2, 3, 0, 0, 0, 7, 6, 8, 4, 5, 5, 0, 5, 1, 2, 7, 4
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OFFSET
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0,1
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LINKS
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FORMULA
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Product_{p odd prime} cos(Pi/p) where Pi = 3.14159...
The log of this constant is equal to Sum_{k>=1} (1 - 2^(2*k))*zeta(2*k)/k * (P(2*k) - 1/2^(2*k)), where P(s) is the prime zeta function. - Amiram Eldar, Aug 21 2020
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EXAMPLE
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cos(Pi/3)*cos(Pi/5)*cos(Pi/7)*cos(Pi/11)*(...) = 0.312832929508881838333...
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MAPLE
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read("transforms") ;
Digits := 300 ;
ZetaM := proc(s, M)
local v, p;
v := Zeta(s) ;
p := 2;
while p <= M do
v := v*(1-1/p^s) ;
p := nextprime(p) ;
end do:
v ;
end proc:
T := 40 ;
preT := 0.0 ;
while true do
cos(Pi/p) ;
subs(p=1/x, %) ;
t := taylor(%, x=0, T) ;
L := [] ;
for i from 1 to T-1 do
L := [op(L), evalf(coeftayl(t, x=0, i))] ;
end do:
Le := EULERi(L) ;
v := 1.0 ;
pre := 0.0 ;
for i from 2 to nops(Le) do
pre := v ;
v := v*evalf(ZetaM(i, 2))^op(i, Le) ;
end do:
pre := (v+pre)/2. ;
printf("%.80f\n", pre) ;
if abs(1.0-preT/pre) < 10^(-Digits/3) then
break;
end if;
preT := pre ;
T := T+15 ;
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MATHEMATICA
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Block[{$MaxExtraPrecision=1000}, Do[Print[Exp[-Sum[N[(2^(2k)-1)*Zeta[2k]/k*(PrimeZetaP[2k]-1/2^(2k)), 120], {k, 1, m}]]], {m, 300, 350}]] (* Vaclav Kotesovec, Jun 02 2015 *)
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PROG
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(PARI) primezeta(n)=sum(k=1, lambertw(10.^default(realprecision)*log(4)) \log(4)+1, moebius(k)*log(zeta(n*k))/k)
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CROSSREFS
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KEYWORD
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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