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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 (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)

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.

LINKS

Table of n, a(n) for n=0..39.

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

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

0.3532363718549959...

CROSSREFS

Sequence in context: A021743 A057023 A245509 * A100481 A205009 A101778

Adjacent sequences:  A085846 A085847 A085848 * A085850 A085851 A085852

KEYWORD

nonn,cons,more

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

STATUS

approved

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Last modified November 28 02:24 EST 2014. Contains 250286 sequences.