OFFSET
2,1
COMMENTS
Theorem (G. Robin): exp(gamma) n log log n - sigma(n) is positive for all n >= 5041 if and only if the Riemann Hypothesis is true.
Note that a(n) <= exp(gamma) n log log n - sigma(n) < a(n) + 1.
REFERENCES
D. S. Mitrinovic et al., Handbook of Number Theory, Kluwer, Section III.2.2.b.
G. Robin, Grandes valeurs de la fonction somme des diviseurs et hypothese de Riemann, J. Math. Pures Appl. 63 (1984), 187-213.
LINKS
T. D. Noe, Table of n, a(n) for n = 2..10000
G. Caveney, J.-L. Nicolas, and J. Sondow, Robin's theorem, primes, and a new elementary reformulation of the Riemann Hypothesis, Integers 11 (2011), #A33.
G. Caveney, J.-L. Nicolas and J. Sondow, On SA, CA, and GA numbers, Ramanujan J., 29 (2012), 359-384.
MAPLE
with(numtheory); Digits := 100; g := evalf(gamma); [seq( floor(exp(g)*n*log(log(n)))-sigma[1](n), n=2..80)];
MATHEMATICA
a[n_] := Floor[Exp[EulerGamma] n*Log[Log[n]]] - DivisorSigma[1, n]; Array[a, 100, 2] (* Jean-François Alcover, May 04 2011 *)
PROG
(PARI) a(n)=floor( exp(Euler)*n*log(log(n)) - sigma(n)) \\ Charles R Greathouse IV, Feb 08 2017
CROSSREFS
KEYWORD
sign,nice,easy
AUTHOR
N. J. A. Sloane, Nov 30 2000
EXTENSIONS
Statement of Robin's theorem corrected by Jonathan Sondow, May 30 2011
STATUS
approved