OFFSET
1,1
COMMENTS
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
REFERENCES
E. C. Titchmarsh and D. R. Heath-Brown, The theory of the Riemann zeta-function, 2nd ed., 1986.
EXAMPLE
2.518249032572967539204071059542687...
MATHEMATICA
RealDigits[N[Exp[Zeta[2]/Exp[EulerGamma]], 115]][[1]]
CROSSREFS
KEYWORD
cons,nonn
AUTHOR
Artur Jasinski, Apr 09 2023
STATUS
approved