A360895 -id:A360895 - cp's OEIS frontend

A362151 Decimal expansion of exp(zeta(2)/exp(gamma)) where gamma is the Euler-Mascheroni constant A001620.

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, 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, 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

Offset: 1

Artur Jasinski, Apr 09 2023

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)).
Proof:
Follow E. C. Titchmarsh and D. R. Heath-Brown p. 67 eq. (3.15.3).
Product_{p<=x} (1+1/p) ~ log(x)*exp(gamma)/zeta(2).
For any particular integer n, it follows from the equations
  n = log(x)*exp(gamma)/zeta(2) -> x_n = exp(n*exp(-gamma)*zeta(2)) and
  n+1 = log(x)*exp(gamma)/zeta(2) -> x_(n+1) = exp((n+1)*exp(-gamma)*zeta(2))
that lim_{n->oo} exp((n+1)*exp(-gamma)*zeta(2))/exp(n*exp(-gamma)*zeta(2)) = exp(zeta(2)/exp(gamma)).
Convergence table:
n        p(n)         truncated product up to p(n)      ratio p(n)/p(n-1)
22     667038311  22.0000000031301736805108740934458   2.51828570030407176
23    1679809291  23.0000000125715665307020553151962   2.51831006300326279
24    4230219377  24.0000000051805484055085694624554   2.51827359192764460
25   10652786759  25.0000000022564574124503565355420   2.51825870235476442
26   26826453991  26.0000000003663337659198715971438   2.51825692167692061
27   67555877849  27.0000000003436918565017475632101   2.51825596747390854
28  170122677001  28.0000000000496255633187331645369   2.51825129681914497
29  428411419313  29.0000000000157769668449867937821   2.51824992919951377
oo            oo  oo                                   2.51824903257296753

			2.518249032572967539204071059542687...

E. C. Titchmarsh and D. R. Heath-Brown, The theory of the Riemann zeta-function, 2nd ed., 1986.

Cf. A001620, A013661, A072997, A360895.

RealDigits[N[Exp[Zeta[2]/Exp[EulerGamma]], 115]][[1]]

Equals exp(A013661/exp(A001620)).

Limit_{n->oo} A072997(n+1)/A072997(n).

cp's OEIS Frontend

A362151 Decimal expansion of exp(zeta(2)/exp(gamma)) where gamma is the Euler-Mascheroni constant A001620.

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