A072113 Continued fraction expansion of Hall and Tenenbaum constant.

0, 3, 23, 1, 1, 16, 1, 2, 1, 8, 1, 274, 3, 1, 5, 1, 2, 1, 16, 1, 3, 3, 2, 1, 4, 1, 1, 2, 1, 2, 1, 2, 1, 1, 2, 16, 3, 3, 2, 1, 1, 1, 2, 69, 121, 1, 5, 1, 2, 1, 2, 1, 1, 1, 2, 1, 12, 4, 1, 1, 1, 1, 2, 1, 2, 3, 3, 1, 3, 2, 4, 1, 7, 1, 16, 2, 4, 1, 2, 7, 2, 3, 1, 3, 2, 1, 1, 1, 1, 2, 1, 1, 3, 1, 1, 3, 2, 1

Offset: 0

Author

Benoit Cloitre, Jun 19 2002

Keywords

• base
• cofr
• easy
• nonn

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).

References

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

Crossrefs

Cf. A072112 (decimal expansion).

Programs

PARI

\p200;
contfrac(cos(solve(X=0,2*Pi,sin(X)+(Pi-X)*cos(X)-Pi/2)))

Formula

K = cos(S) = 0.3287... where S it the root 0< S < 2Pi of sin(S)+(Pi-S)*cos(S) = Pi/2.

Extensions

Offset changed by Andrew Howroyd, Jul 06 2024