A072113 -id:A072113 - cp's OEIS frontend

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

Benoit Cloitre, Jun 19 2002

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

			0.32867416290854621681828451404311511897694154765578...

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

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.

Cf. A072113.

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 *)

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

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

cp's OEIS Frontend

A072112 Decimal expansion of Hall and Tenenbaum constant.

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