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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
 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 (list; constant; graph; refs; listen; history; text; internal format)
 OFFSET 0,1 COMMENTS The Hafner-Sarnak-McCurley constant. 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) LINKS Ph. Flajolet and I. Vardi, Zeta function expansions of some classical constants J. L. Hafner, P. Sarnak and K. McCurley, Relatively prime values of polynomials, In A Tribute to Emil Grosswald: Number Theory and Related Analysis, Contemporary Mathematics (1993), M. Knopp and M. Sheigorn, Editors, vol. 143. Eric Weisstein's World of Mathematics, Hafner-Sarnak-McCurley Constant Wikipedia, Hafner-Sarnak-McCurley constant FORMULA s = Prod(1-(1-prod(n>=1, 1-1/p^n))^2) where p runs through the primes; s=0.35323637185499598454... - 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 Sequence in context: A021743 A057023 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

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Last modified August 20 15:13 EDT 2019. Contains 326152 sequences. (Running on oeis4.)