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Consider the set of polynomials (in standard form) P: the coefficients in the polynomials are integers.

Quite like Z (integers), every distinct member of P has a unique 'prime factorization'. (Although all 1st degree or higher polynomials always have factors within C (complex polynomials), they seemingly rarely have factors within P)

Then some members of P are coprime to others.

What is the probability that n randomly chosen members of P, degree z or lower, are coprime?

This is already solved for z=0, due to Eulers solution to the Basel problem and the Zeta function. This quandary seems a lot more difficult because I don't know of a way to generate prime polynomials, etc.

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3  
What sequence of finite subsets do you use to define a randomly chosen member of P of degree z or lower? –  Ricky Demer Oct 29 '10 at 6:01
    
@Ricky: I guess, since he refers to Euler, he takes all pairs of polynomials with coefficients between $-n$ and $n$, $n=1,2,...$. Is there a more natural way? –  Mark Sapir Oct 29 '10 at 10:03
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Most polynomials are irreducible over the rational so, ignoring constant factors, your probability is zero. –  Felipe Voloch Oct 29 '10 at 11:48
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Felipe, OP asks the probability of coprimality, so I think you mean one, not zero. –  Gerry Myerson Oct 29 '10 at 12:20
    
Thank you, Gerry. –  Felipe Voloch Oct 29 '10 at 12:23
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