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A233825 Decimal expansion of Nicolas's constant in his condition for the Riemann Hypothesis (RH). 3
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 (list; constant; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
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).
LINKS
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.
FORMULA
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.
EXAMPLE
3.64441509640737014106511619283514816005226024664324245685246375826374...
MATHEMATICA
RealDigits[Exp[EulerGamma]*(4 + EulerGamma - Log[4*Pi]), 10, 120][[1]] (* Amiram Eldar, May 25 2023 *)
PROG
(PARI) exp(Euler)*(4 + Euler - log(4*Pi)) \\ Charles R Greathouse IV, Mar 10 2016
CROSSREFS
Sequence in context: A090038 A308291 A006464 * A351124 A159354 A196500
KEYWORD
nonn,cons
AUTHOR
Jonathan Sondow, Dec 19 2013
STATUS
approved

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Last modified April 21 00:22 EDT 2024. Contains 371850 sequences. (Running on oeis4.)