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A085849 Decimal expansion of the probability that two m X m and n X n matrices (m,n large) have relatively prime determinants. 0

%I #37 Jun 15 2021 07:30:55

%S 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,

%T 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,

%U 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

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

%C 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]

%C Comment on numerics from _R. J. Mathar_, Apr 20 2011: (Start)

%C 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.

%C 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.

%C (End)

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

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

%H Philippe Flajolet and Ilan Vardi, <a href="http://algo.inria.fr/flajolet/Publications/landau.ps">Zeta function expansions of some classical constants</a>.

%H J. L. Hafner, P. Sarnak and K. McCurley, <a href="http://www.mccurley.org/papers/relprime.pdf">Relatively prime values of polynomials</a>, in: M. Knopp and M. Sheigorn, Editors, A Tribute to Emil Grosswald: Number Theory and Related Analysis, Contemporary Mathematics, Vol. 143, AMS, 1993.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Hafner-Sarnak-McCurleyConstant.html">Hafner-Sarnak-McCurley Constant</a>.

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Hafner-Sarnak-McCurley_constant">Hafner-Sarnak-McCurley constant</a>.

%F Equals Product_{p prime} (1-(1-Product_{n>=1} (1-1/p^n))^2). - _Benoit Cloitre_, Aug 05 2003

%e 0.3532363718549959845435165504326820112801647785666904464160859428...

%t 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 *)

%Y Cf. A002107, A010815.

%K nonn,cons

%O 0,1

%A _Eric W. Weisstein_, Jul 05 2003

%E More terms from _Benoit Cloitre_, Aug 05 2003

%E Edited by _N. J. A. Sloane_, Feb 11 2009 at the suggestion of _R. J. Mathar_

%E Twenty additional digits from _R. J. Mathar_, Feb 13 2009

%E Extended to 100 digits by _Jean-François Alcover_, Apr 25 2016

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Last modified March 18 22:56 EDT 2024. Contains 370952 sequences. (Running on oeis4.)