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

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

Author

Jonathan Sondow, Oct 24 2012

Keywords

• nonn

Comments

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.

Examples

			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.
		

Crossrefs

Cf. A216868, A233825.

Programs

Mathematica

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}]

Formula

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].