A218245 -id:A218245 - cp's OEIS frontend

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

Jonathan Sondow, Sep 29 2012

nonn

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

			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.

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.

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.

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

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

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

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

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.

A233825 Decimal expansion of Nicolas's constant in his condition for the Riemann Hypothesis (RH).

Original entry on oeis.org

3, 6, 4, 4, 4, 1, 5, 0, 9, 6, 4, 0, 7, 3, 7, 0, 1, 4, 1, 0, 6, 5, 1, 1, 6, 1, 9, 2, 8, 3, 5, 1, 4, 8, 1, 6, 0, 0, 5, 2, 2, 6, 0, 2, 4, 6, 6, 4, 3, 2, 4, 2, 4, 5, 6, 8, 5, 2, 4, 6, 3, 7, 5, 8, 2, 6, 3, 7, 4, 1, 7, 3, 4, 8, 0, 9, 2, 9, 5, 8, 1, 8, 6, 8, 3, 2, 3, 0, 5, 7, 0, 5, 1, 7, 5, 1, 2, 6, 1, 6, 1, 5, 5, 6, 4, 1, 4, 3, 3, 5, 5, 3, 1, 7, 7, 5, 2, 9, 2, 7

Offset: 1

Jonathan Sondow, Dec 19 2013

Nicolas proved that RH is true if and only if limsup_{n-->infinity} (n/phi(n) - e^gamma*log(log(n)))*sqrt(log(n)) = e^gamma*(4 + gamma - log(4*Pi)), where phi(n) = A000010(n).

			3.64441509640737014106511619283514816005226024664324245685246375826374...

Jeffrey C. Lagarias, Euler's constant: Euler's work and modern developments, Bull. A.M.S., 50 (2013), 527-628; see p. 574.
Jean-Louis Nicolas, Small values of the Euler function and the Riemann hypothesis, Acta Arith., Vol. 155, No. 3 (2012), pp. 311-321; arXiv preprint, arXiv:1202.0729 [math.NT], 2012.

Cf. A000010, A001620, A195423, A216868, A218245.

RealDigits[Exp[EulerGamma]*(4 + EulerGamma - Log[4*Pi]), 10, 120][[1]] (* Amiram Eldar, May 25 2023 *)

exp(Euler)*(4 + Euler - log(4*Pi)) \\ Charles R Greathouse IV, Mar 10 2016

Equals e^gamma*(4 + gamma - log(4*Pi)), where gamma is the Euler-Mascheroni constant.

Equals e^gamma*(2 + beta), where beta = Sum 1/(rho*(1-rho)), where rho runs over all nonreal zeros of the zeta function.

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

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A233825 Decimal expansion of Nicolas's constant in his condition for the Riemann Hypothesis (RH).

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