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A216868 Nicolas's sequence whose positivity is equivalent to the Riemann hypothesis. 3
3, 4, 13, 67, 560, 6095, 87693, 1491707, 30942952, 795721368, 22614834943, 759296069174, 28510284114397, 1148788714239052, 50932190960133487, 2532582753383324327, 139681393339880282191, 8089483267352888074399, 512986500081861276401709, 34658318003703434434962860 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

a(n) = p(n)# - [phi(p(n)#)*log(log(p(n)#))*exp(gamma)], where p(n)# is the n-th primorial, phi is Euler's totient function, gamma is Euler's constant, and [.] denotes the floor function.

All a(n) are > 0 if and only if the Riemann hypothesis is true. If the Riemann hypothesis is false, then infinitely many a(n) are > 0 and infinitely many a(n) are <= 0. Nicolas (1983) proved this with a(n) replaced by p(n)#/phi(p(n)#)-log(log(p(n)#))*exp(gamma). Nicolas's refinement of this result is in A233825.

See A185339 for additional links, references, and formulas.

REFERENCES

J.-L. Nicolas, Petites valeurs de la fonction d'Euler et hypothese de Riemann, in Seminar on Number Theory, Paris 1981-82 (Paris 1981/1982), Birkhauser, Boston, 1983, pp. 207-218.

LINKS

Table of n, a(n) for n=1..20.

J.-L. Nicolas, Petites valeurs de la fonction d'Euler,  J. Number Theory, 17 no.3 (1983), 375-388.

J.-L. Nicolas, Small values of the Euler function and the Riemann hypothesis, Acta Arith., 155 (2012), 311-321.

FORMULA

a(n) = p(n)# - [phi(p(n)#)*log(log(p(n)#))*e^gamma].

a(n) = A002110(n) - [A005867(n)*log(log(A002110(n)))*e^gamma].

lim(n->infty, a(n)/p(n)#) = 0.

EXAMPLE

p(2)# = 2*3 = 6 and phi(6) = 2, so a(2) = 6 - [2*log(log(6))*e^gamma] = 6 - [2*0.58319...*1.78107...] = 6 - [2.07...] = 6 - 2 = 4.

MATHEMATICA

primorial[n_] := Product[Prime[k], {k, n}]; Table[With[{p = primorial[n]}, p - Floor[EulerPhi[p]*Log[Log[p]]*Exp[EulerGamma]]], {n, 1, 20}]

PROG

(PARI) nicolas(n) = {p = 2; pri = 2; for (i=1, n, print1(pri - floor(eulerphi(pri)*log(log(pri))*exp(Euler)), ", "); p = nextprime(p+1); pri *= p; ); } \\ Michel Marcus, Oct 06 2012

(PARI) A216868(n)={(n=prod(i=1, n, prime(i)))-floor(eulerphi(n)*log(log(n))*exp(Euler))}  \\ M. F. Hasler, Oct 06 2012

CROSSREFS

Cf. A000010, A001620, A002110, A005867, A185339, A209079, A218245, A233825.

Sequence in context: A001056 A122151 A294384 * A082732 A220846 A009286

Adjacent sequences:  A216865 A216866 A216867 * A216869 A216870 A216871

KEYWORD

nonn

AUTHOR

Jonathan Sondow, Sep 29 2012

STATUS

approved

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Last modified October 23 18:52 EDT 2018. Contains 316530 sequences. (Running on oeis4.)