A085849 Decimal expansion of the probability that two m X m and n X n matrices (m,n large) have relatively prime determinants.

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

Offset

0,1

Comments

The Hafner-Sarnak-McCurley constant. [Named after the American mathematician James Lee Hafner (1954-2015), the South-African and American mathematician Peter Clive Sarnak (b. 1953) and the American mathematician and computer scientist Kevin Snow McCurley. - Amiram Eldar, Jun 15 2021]

Comment on numerics from R. J. Mathar, Apr 20 2011: (Start)

The definition s = Product_{p} (1-[1- Product_{n>=1} (1-1/p^n)]^2) may be binomially expanded to s = Product_{p} Sum_{n>=1} (2*A010815(n)-A002107(n))/p^n. The auxiliary sequence 2*A010815(n)-A002107(n) is 1, 0, -1, -2, -1, 0, 2, 2, 2, 2, -1, 0,... for n>=0.

The inverse Euler transformation of the auxiliary sequence generates Sum_{n} (2*A010815(n)-A002107(n)) /p^n = Product_{n} (1-1/p^n)^gamma(n) with gamma(n) = 0, -1, -2 ,-1, -2, 0, -2, -1, 0, -2, 0, -1,... for n>=1. This yields s = Product_{n>=1} zeta(n)^gamma(n) where zeta(n) are the values of the Riemann zeta function.

(End)

References

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, section 2.5, "Hafner-Sarnak-McCurley Constant", pp. 110-112.

Ilan Vardi, Computational Recreations in Mathematica, Redwood City, CA: Addison-Wesley, 1991, p. 174.

Links

Table of n, a(n) for n=0..101.

Philippe Flajolet and Ilan Vardi, Zeta function expansions of some classical constants.

J. L. Hafner, P. Sarnak and K. McCurley, Relatively prime values of polynomials, in: M. Knopp and M. Sheigorn, Editors, A Tribute to Emil Grosswald: Number Theory and Related Analysis, Contemporary Mathematics, Vol. 143, AMS, 1993.

Eric Weisstein's World of Mathematics, Hafner-Sarnak-McCurley Constant.

Wikipedia, Hafner-Sarnak-McCurley constant.

Formula

Equals Product_{p prime} (1-(1-Product_{n>=1} (1-1/p^n))^2). - Benoit Cloitre, Aug 05 2003

Example

0.3532363718549959845435165504326820112801647785666904464160859428...

Mathematica

digits = 102; CC = CoefficientList[Log[1 - (1 - QPochhammer[1/p])^2] + O[p, Infinity]^(4 digits), 1/p][[3 ;; -1]]; Hafner = CC.Table[PrimeZetaP[n + 1], {n, 1, Length[CC]}] // Exp // N[#, digits+10]&; RealDigits[Hafner, 10, digits][[1]] (* Jean-François Alcover, Apr 25 2016 *)

Crossrefs

Cf. A002107, A010815.

Sequence in context: A332438 A357469 A245509 * A100481 A205009 A101778

Adjacent sequences: A085846 A085847 A085848 * A085850 A085851 A085852

Keyword

nonn,cons

Author

Eric W. Weisstein, Jul 05 2003

Extensions

More terms from Benoit Cloitre, Aug 05 2003

Edited by N. J. A. Sloane, Feb 11 2009 at the suggestion of R. J. Mathar

Twenty additional digits from R. J. Mathar, Feb 13 2009

Extended to 100 digits by Jean-François Alcover, Apr 25 2016

Status

approved