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%I #36 Jan 04 2021 14:10:11
%S 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,
%T 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,
%U 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
%N Decimal expansion of prime analog of the Kepler-Bouwkamp constant: Product_{k>=2} cos(Pi/prime(k)).
%H Vaclav Kotesovec, <a href="/A131671/b131671.txt">Table of n, a(n) for n = 0..1000</a>
%H Steven R. Finch, <a href="http://arxiv.org/abs/2001.00578">Errata and Addenda to Mathematical Constants</a>, arXiv:2001.00578 [math.HO], 2020, p. 62.
%H Adrian R. Kitson, <a href="https://www.jstor.org/stable/27821786">The prime analog of the Kepler-Bouwkamp constant</a>, The Mathematical Gazette, Vol. 92, No. 293 (2008), pp. 293-295, <a href="http://arxiv.org/abs/math.HO/0608186">preprint</a>, arXiv:math/0608186 [math.HO], 2006.
%H R. J. Mathar, <a href="http://arxiv.org/abs/1301.6293">Tightly circumscribed regular polygons</a>, arXiv:1301.6293 [math.MG], 2013.
%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Kepler-Bouwkamp_constant">Kepler-Bouwkamp constant</a>
%F Product_{p odd prime} cos(Pi/p) where Pi = 3.14159...
%F 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
%e cos(Pi/3)*cos(Pi/5)*cos(Pi/7)*cos(Pi/11)*(...) = 0.312832929508881838333...
%p read("transforms") ;
%p Digits := 300 ;
%p ZetaM := proc(s,M)
%p local v,p;
%p v := Zeta(s) ;
%p p := 2;
%p while p <= M do
%p v := v*(1-1/p^s) ;
%p p := nextprime(p) ;
%p end do:
%p v ;
%p end proc:
%p T := 40 ;
%p preT := 0.0 ;
%p while true do
%p cos(Pi/p) ;
%p subs(p=1/x,%) ;
%p t := taylor(%,x=0,T) ;
%p L := [] ;
%p for i from 1 to T-1 do
%p L := [op(L),evalf(coeftayl(t,x=0,i))] ;
%p end do:
%p Le := EULERi(L) ;
%p v := 1.0 ;
%p pre := 0.0 ;
%p for i from 2 to nops(Le) do
%p pre := v ;
%p v := v*evalf(ZetaM(i,2))^op(i,Le) ;
%p end do:
%p pre := (v+pre)/2. ;
%p printf("%.80f\n",pre) ;
%p if abs(1.0-preT/pre) < 10^(-Digits/3) then
%p break;
%p end if;
%p preT := pre ;
%p T := T+15 ;
%p end do: # _R. J. Mathar_, Jan 23 2013
%t 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 *)
%o (PARI) primezeta(n)=sum(k=1, lambertw(10.^default(realprecision)*log(4)) \log(4)+1, moebius(k)*log(zeta(n*k))/k)
%o exp(-suminf(k=1,(4^k-1)*zeta(2*k)/k*(primezeta(2*k)-1/4^k))) \\ _M. F. Hasler_ and _Charles R Greathouse IV_, May 28 2015
%Y Cf. A085365.
%K cons,nonn
%O 0,1
%A _R. J. Mathar_, Sep 12 2007
%E More digits from _R. J. Mathar_, Mar 01 2009, Jan 23 2013
%E Edited by _M. F. Hasler_, May 18 2014
%E More digits from _Vaclav Kotesovec_, Jun 02 2015
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