A216868 Nicolas's sequence whose positivity is equivalent to the Riemann hypothesis.

3, 4, 13, 67, 560, 6095, 87693, 1491707, 30942952, 795721368, 22614834943, 759296069174, 28510284114397, 1148788714239052, 50932190960133487, 2532582753383324327, 139681393339880282191, 8089483267352888074399, 512986500081861276401709, 34658318003703434434962860

Offset

1,1

Comments

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

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.

Named after the French mathematician Jean-Louis Nicolas. - Amiram Eldar, Jun 23 2021

References

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

Links

Amiram Eldar, Table of n, a(n) for n = 1..350

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

J.-L. Nicolas, Small values of the Euler function and the Riemann hypothesis, arXiv:1202.0729 [math.NT], 2012; Acta Arith., Vol. 155 (2012), pp. 311-321.

Formula

a(n) = prime(n)# - floor(phi(prime(n)#)*log(log(prime(n)#))*e^gamma).

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

Limit_{n->oo} a(n)/p(n)# = 0.

Example

prime(2)# = 2*3 = 6 and phi(6) = 2, so a(2) = 6 - floor(2*log(log(6))*e^gamma) = 6 - floor(2*0.58319...*1.78107...) = 6 - floor(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 A307893 A337297

Adjacent sequences: A216865 A216866 A216867 * A216869 A216870 A216871

Keyword

nonn

Author

Jonathan Sondow, Sep 29 2012

Status

approved