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%I #31 Jun 13 2021 05:13:04
%S 6,7,8,2,3,4,4,9,1,9,1,7,3,9,1,9,7,8,0,3,5,5,3,8,2,7,9,4,8,2,8,9,4,8,
%T 1,4,0,9,6,3,3,2,2,3,9,1,8,9,4,4,0,1,0,3,0,3,6,4,6,0,4,1,5,9,6,4,9,8,
%U 3,3,7,0,7,4,0,1,2,3,2,3,3,2,1,3,7,6,2,1,2,2,9,3,3,4,8,4,6,1,6,8,8,8,3,2,8
%N Decimal expansion of Product_{p prime} (1-3/p^3+2/p^4+1/p^5-1/p^6).
%C Equals (49/64)*(668/729)*(15304/15625)*(116724/117649)*... inserting p= A000040 = 2, 3, 5, 7.. into the factor. Slightly larger than Product_{p=primes} (1-3/p^3) = 0.534566872085103888416775...
%H Steven R. Finch, <a href="/A000924/a000924.pdf">Class number theory</a>, 2005. [Cached copy, with permission of the author]
%H Steven R. Finch, <a href="https://doi.org/10.1017/9781316997741">Mathematical Constants II</a>, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, 2018, p. 85.
%H Takashi Taniguchi, <a href="https://aif.centre-mersenne.org/item/?id=AIF_2008__58_2_625_0">A mean value theorem for the square of class number times regulator of quadratic extensions</a>, Annales de l'institut Fourier. Vol. 58, No. 2 (2008), pp. 625-670; <a href="https://arxiv.org/abs/math/0410531">arXiv preprint</a>, arXiv:math/0410531 [math.NT], 2004-2006.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PrimeProducts.html">Prime Products</a>.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/TaniguchisConstant.html">Taniguchi's Constant</a>.
%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Euler_product">Euler Product</a>.
%e 0.678234491917391978035...
%p read("transforms") : efact := 1-3/p^3+2/p^4+1/p^5-1/p^6 ; Digits := 130 : tm := 310 : subs (p=1/x,1/efact) ; taylor(%,x=0,tm) : L := [seq(coeftayl(%,x=0,i),i=1..tm-1)] : Le := EULERi(L) : x := 1.0 :
%p for i from 2 to nops(Le) do x := x/evalf(Zeta(i))^op(i,Le) ; x := evalf(x) ; print(x) ; end do:
%t digits = 105; $MaxExtraPrecision = 400; m0 = 1000; dm = 100; Clear[s];
%t LR = LinearRecurrence[{0, 0, 3, -2, -1, 1}, {0, 0, -9, 8, 5, -33}, 2 m0];
%t r[n_Integer] := LR[[n]]; s[m_] := s[m] = NSum[r[n] PrimeZetaP[n]/n, {n, 3, m}, NSumTerms -> m0, WorkingPrecision -> 400] // Exp; s[m0]; s[m = m0 + dm]; While[RealDigits[s[m], 10, digits][[1]] != RealDigits[s[m-dm], 10, digits][[1]], Print[m]; m = m+dm]; RealDigits[s[m], 10, digits][[1]] (* _Jean-François Alcover_, Apr 15 2016 *)
%o (PARI) prodeulerrat(1-3/p^3+2/p^4+1/p^5-1/p^6) \\ _Amiram Eldar_, Mar 04 2021
%Y Cf. A256392, A330523.
%K cons,nonn
%O 0,1
%A _R. J. Mathar_, Aug 01 2010
%E More digits from _Jean-François Alcover_, Apr 15 2016
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