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%I #20 May 12 2023 16:28:41
%S 2,5,1,8,2,4,9,0,3,2,5,7,2,9,6,7,5,3,9,2,0,4,0,7,1,0,5,9,5,4,2,6,8,7,
%T 0,0,2,9,3,4,5,3,5,8,6,7,8,8,6,7,7,9,4,3,8,1,4,6,2,0,6,2,1,8,5,3,6,8,
%U 3,9,3,9,9,3,4,8,8,4,6,9,4,2,4,9,3,5,1,6,9,3,4,0,5,8,5,4,2,0,8,9,8,5,0,6,8,0,4,4,0,4,2,1,8,7,3
%N Decimal expansion of exp(zeta(2)/exp(gamma)) where gamma is the Euler-Mascheroni constant A001620.
%C Theorem: Let p(n) be the smallest prime p such that Product_{prime p<=p(n)} (1 + 1/p) >= n. Then lim_{n->oo} p(n+1)/p(n) = exp(zeta(2)/exp(gamma)).
%C Proof:
%C Follow E. C. Titchmarsh and D. R. Heath-Brown p. 67 eq. (3.15.3).
%C Product_{p<=x} (1+1/p) ~ log(x)*exp(gamma)/zeta(2).
%C For any particular integer n, it follows from the equations
%C n = log(x)*exp(gamma)/zeta(2) -> x_n = exp(n*exp(-gamma)*zeta(2)) and
%C n+1 = log(x)*exp(gamma)/zeta(2) -> x_(n+1) = exp((n+1)*exp(-gamma)*zeta(2))
%C that lim_{n->oo} exp((n+1)*exp(-gamma)*zeta(2))/exp(n*exp(-gamma)*zeta(2)) = exp(zeta(2)/exp(gamma)).
%C Convergence table:
%C n p(n) truncated product up to p(n) ratio p(n)/p(n-1)
%C 22 667038311 22.0000000031301736805108740934458 2.51828570030407176
%C 23 1679809291 23.0000000125715665307020553151962 2.51831006300326279
%C 24 4230219377 24.0000000051805484055085694624554 2.51827359192764460
%C 25 10652786759 25.0000000022564574124503565355420 2.51825870235476442
%C 26 26826453991 26.0000000003663337659198715971438 2.51825692167692061
%C 27 67555877849 27.0000000003436918565017475632101 2.51825596747390854
%C 28 170122677001 28.0000000000496255633187331645369 2.51825129681914497
%C 29 428411419313 29.0000000000157769668449867937821 2.51824992919951377
%C oo oo oo 2.51824903257296753
%D E. C. Titchmarsh and D. R. Heath-Brown, The theory of the Riemann zeta-function, 2nd ed., 1986.
%F Equals exp(A013661/exp(A001620)).
%F Limit_{n->oo} A072997(n+1)/A072997(n).
%e 2.518249032572967539204071059542687...
%t RealDigits[N[Exp[Zeta[2]/Exp[EulerGamma]], 115]][[1]]
%Y Cf. A001620, A013661, A072997, A360895.
%K cons,nonn
%O 1,1
%A _Artur Jasinski_, Apr 09 2023