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A085849
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Decimal expansion of the probability that two m X m and n X n matrices (m,n large) have relatively prime determinants.
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0
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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
(list; constant; graph; refs; listen; history; internal format)
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OFFSET
| 0,1
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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)
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REFERENCES
| P. Flajolet and Ilan Vardi, Zeta function Expansions of Classical constants, Feb. 18, 1996
Hafner, J. L., Sarnak, P. and McCurley, K., Relatively prime values of polynomials, In contemporary Mathematics (1993), M. Knopp and M. Sheigorn, Editors, vol. 143.
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LINKS
| Ph. Flajolet and I. Vardi, Zeta function expansions of some classical constants
Eric Weisstein's World of Mathematics, Hafner-Sarnak-McCurley Constant
Wikipedia, Hafner-Sarnak-McCurley constant
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FORMULA
| s=prod(1-(1-prod(n>=1, 1-1/p^n))^2) where p runs through the primes; s=0.35323637185499598454... - Benoit Cloitre (benoit7848c(AT)orange.fr), Aug 05 2003
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EXAMPLE
| 0.3532363718549959...
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CROSSREFS
| Sequence in context: A196997 A021743 A057023 * A100481 A205009 A101778
Adjacent sequences: A085846 A085847 A085848 * A085850 A085851 A085852
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KEYWORD
| nonn,cons,more
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AUTHOR
| Eric Weisstein (eric(AT)weisstein.com), Jul 05, 2003
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EXTENSIONS
| More terms from Benoit Cloitre (benoit7848c(AT)orange.fr), Aug 05 2003
Edited by N. J. A. Sloane (njas(AT)research.att.com), Feb 11 2009 at the suggestion of R. J. Mathar
Twenty additional digits from R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Feb 13 2009
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