A072112 Decimal expansion of Hall and Tenenbaum constant.

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

Offset

0,1

Comments

For any multiplicative function g with values -1<= g(k) <= 1, for any real x >=2, Sum( i<= x, g(i) ) << x * exp{ -K * Sum( p<=x, (1-g(p))/p ) } and K is the optimal constant satisfying this inequality (Hall and Tenenbaum, 1991).

Named after the British mathematician Richard Roxby Hall and the French mathematician Gérald Tenenbaum (b. 1952). - Amiram Eldar, Jun 22 2021

References

G. Tenenbaum, Introduction à la théorie analytique et probabiliste des nombres, p. 348, Publications de l'Institut Cartan, 1990.

Links

Table of n, a(n) for n=0..103.

R. R. Hall and G. Tenenbaum, Effective mean value estimates for complex multiplicative functions, Mathematical Proceedings of the Cambridge Philosophical Society, Vol. 110, No. 2 (1991), pp. 337-351.

Formula

K = cos(S) = 0.3286... where S is the root 0 < S < 2*Pi of sin(S)+(Pi-S)*cos(S) = Pi/2.

Example

0.32867416290854621681828451404311511897694154765578...

Mathematica

digits = 104; x /. FindRoot[Pi*x + Sqrt[1 - x^2] - x*ArcCos[x] == Pi/2, {x, 0}, WorkingPrecision -> digits] // RealDigits[#, 10, digits]& // First (* Jean-François Alcover, Feb 15 2013 *)

Prog

(PARI) \p 200;
cos(solve(X=0, 2*Pi, sin(X)+(Pi-X)*cos(X)-Pi/2))

Crossrefs

Cf. A072113.

Sequence in context: A072657 A098163 A260323 * A071658 A089860 A130960

Adjacent sequences: A072109 A072110 A072111 * A072113 A072114 A072115

Keyword

cons,easy,nonn

Author

Benoit Cloitre, Jun 19 2002

Status

approved