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A216868 Nicolas's sequence whose positivity is equivalent to the Riemann hypothesis. 3

%I

%S 3,4,13,67,560,6095,87693,1491707,30942952,795721368,22614834943,

%T 759296069174,28510284114397,1148788714239052,50932190960133487,

%U 2532582753383324327,139681393339880282191,8089483267352888074399,512986500081861276401709,34658318003703434434962860

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

%C 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.

%C 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.

%C See A185339 for additional links, references, and formulas.

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

%H Amiram Eldar, <a href="/A216868/b216868.txt">Table of n, a(n) for n = 1..350</a>

%H J.-L. Nicolas, <a href="http://math.univ-lyon1.fr/~nicolas/petitsphi83.pdf">Petites valeurs de la fonction d'Euler</a>, J. Number Theory, 17 no.3 (1983), 375-388.

%H J.-L. Nicolas, <a href="http://arxiv.org/abs/1202.0729">Small values of the Euler function and the Riemann hypothesis</a>, Acta Arith., 155 (2012), 311-321.

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

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

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

%e 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.

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

%o (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

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

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

%K nonn

%O 1,1

%A _Jonathan Sondow_, Sep 29 2012

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Last modified April 10 02:39 EDT 2020. Contains 333392 sequences. (Running on oeis4.)