A088453 Decimal expansion of 1/zeta(3).

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

Offset: 0

Author

Eric W. Weisstein, Sep 30 2003

Keywords

• nonn
• cons
• easy
• changed

Comments

This is the probability that three randomly chosen integers are relatively prime (see A018805). - Gary McGuire, Dec 13 2004

This is also the probability that a random integer is cubefree. - Eugene Salamin, Dec 13 2004

On the other hand, the probability that three randomly-chosen integers are pairwise relatively prime is given by A065473. - Charles R Greathouse IV, Nov 14 2011

This is also the 'probability' that a random algebraic number's denominator is equal to its leading coefficient, see Arno, Robinson, & Wheeler. - Charles R Greathouse IV, Nov 12 2014

This is the probability that a random point on a cubic lattice is visible from the origin, i.e., there is no other lattice point that lies on the line segment between this point and the origin. - Amiram Eldar, Jul 08 2020

Examples

			0.831907372580707468683126278821530734417...
		

References

• Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 1.6, p. 41.
• David Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, Revised edition 1987, p. 29.

Links

• Steven Arno, M. L. Robinson, and Ferell S. Wheeler, On denominators of algebraic numbers and integer polynomials, Journal of Number Theory 57:2 (April 1996), pp. 292-302.
• Eric Weisstein's World of Mathematics, Relatively Prime.

Crossrefs

Cf. A002117, A008683.

Programs

Mathematica

RealDigits[1/Zeta[3],10,120][[1]] (* Harvey P. Dale, May 31 2019 *)

Maxima

fpprec : 200$ bfloat( 1/zeta(3))$ bfloat(%); /* Martin Ettl, Oct 15 2012 */

PARI

1/zeta(3) \\ Charles R Greathouse IV, Nov 12 2014

Formula

Equals 1/A002117.

From Amiram Eldar, Aug 20 2020: (Start)

Equals Sum_{k>=1} mu(k)/k^3, where mu is the Möbius function (A008683).

Equals Product_{p prime} (1 - 1/p^3). (End)

Extensions

Entry revised by N. J. A. Sloane, Dec 16 2004