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Bateman & Horn [1], building on Bateman & Stemmler [2], give a conjectured formula for the density of numbers that produce simultaneous primes in a number of fixed polynomials.

The constant is

$$C=\prod_p\left\{\left(1-\frac1p\right)^{-k}\left(1-\frac{\omega(p)}{p}\right)\right\}$$

where $k$ is the number of polynomials and $\omega(p)$ is the number of residues (mod p) where at least one of the polynomials is divisible by $p$.

Is there a good way to calculate this constant for a given set of polynomials? I'm looking at polynomials with 'reasonable' degrees and constants, nothing like the recordbreaking monsters considered by (e.g.) Jacobson & Williams. Also, I don't need high precision (a dozen digits would be great), but of course direct computation is out of the question as the product converges too slowly to get even three or four digits of precision in a reasonable time.

Related question: Calculating the infinite product from the Hardy-Littlewood Conjecture F

[1] Paul T. Bateman and Roger A. Horn, "A heuristic asymptotic formula concerning the distribution of prime numbers", Mathematics of Computation 16:79 (1962), pp. 363-367.

[2] Paul T. Bateman and Rosemarie M. Stemmler, "Waring's problem in algebraic number fields and primes of the form $(p^r - 1)/(p^d-1)$", Illinois Journal of Mathematics 6 (1962), pp. 142-156.

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1 Answer 1

up vote 1 down vote accepted

See Pascal Sebah and Xavier Gourdon, Constants from number theory, available at http://numbers.computation.free.fr/Constants/Miscellaneous/constantsNumTheory.html

If the formulas are unreadable with your software, as they are with mine, click on the link for the Postscript version, which looks fine on my machine.

EDIT: Similar constants are discussed in 2.1 of Finch, Mathematical Constants, especially pp. 89-91. Among the references given are Shanks, Math Comp 14 (1960) 201-203; Lal, Math Comp 21 (1967) 245-247; Shanks, Math Comp 21 (1967) 705-707.

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I've read the paper, but I don't see how it applies. –  Charles Apr 6 '11 at 15:15
    
@Charles, maybe it doesn't. They do talk about computing $\prod(1-p^{-1}(p-1)^{-1})$, $\prod(1-(p-1)^{-2})$, $\prod(1-(f(p)/g(p)))$ where $f$ and $g$ are polynomials with integer coefficients, and I imagined some of the methods (and some of the references) would be applicable to the products you're looking at. –  Gerry Myerson Apr 7 '11 at 0:52
    
OK, thanks, I'll look into that. –  Charles Apr 11 '11 at 19:41

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