A233825 -id:A233825 - cp's OEIS frontend

A195423 Decimal expansion of -2B = sum(r in Z, 1/(r*(1-r))), where Z is the set of zeros of the Riemann zeta function which lie in the strip 0 <= Re(z) <= 1.

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

Offset: 0

Jonathan Sondow, Sep 18 2011

See A074760 for references and links.

			-2B = gamma + 2 - log(4*Pi) = 0.046191417932242...

Cf. A002410, A074760, A233825.

RealDigits[ N[ EulerGamma + 2 - Log[4*Pi], 105], 10, 100] [[1]]

Euler-log(4*Pi)+2 \\ Charles R Greathouse IV, Mar 10 2016

As a constant, equals 2*A074760.

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

Original entry on oeis.org

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.

A218245 Nicolas's sequence, whose nonnegativity is equivalent to the Riemann hypothesis.

Original entry on oeis.org

2, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 1

Jonathan Sondow, Oct 24 2012

nonn

a(n) = floor(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, and gamma is Euler's constant.
J.-L. Nicolas proved that all terms are >= 0 if and only if the Riemann hypothesis (RH) is true. In fact, results in his 2012 paper imply that RH is equivalent to a(n) = 0 for n > 6. Nicolas's refinement of this result is in A233825.
He also proved that if RH is false, then infinitely many terms are >= 0 and infinitely many terms are < 0.
See Nicolas's sequence A216868 for references, links, and additional cross-refs.

			p(2)# = 2*3 = 6 and phi(6) = 2, so a(2) = [6/2 - log(log(6))*e^gamma] = [3-0.58319...*1.78107...] = [3-1.038...] = 1.

Cf. A216868, A233825.

primorial[n_] := Product[Prime[k], {k, n}]; Table[ With[{p = primorial[n]}, Floor[N[p/EulerPhi[p] - Log[Log[p]]*Exp[EulerGamma]]]], {n, 1, 100}]

a(n) = [p(n)#/phi(p(n)#) - log(log(p(n)#))*exp(gamma)].

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

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A195423 Decimal expansion of -2B = sum(r in Z, 1/(r*(1-r))), where Z is the set of zeros of the Riemann zeta function which lie in the strip 0 <= Re(z) <= 1.

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

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A218245 Nicolas's sequence, whose nonnegativity is equivalent to the Riemann hypothesis.

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