Re: Density of sequence questions
- From: Gerry Myerson <gerry@xxxxxxxxxxxxxxxxxxxxxxxxx>
- Date: Tue, 07 Jul 2009 15:36:45 +1000
In article
<19599177.78181.1246866592606.JavaMail.jakarta@xxxxxxxxxxxxxxxxxxxxxx>,
"Richard L. Peterson" <rl_pete@xxxxxxxxx> wrote:
I asked:
I should have asked also:If q is cofinite in P,Gerry Myerson replied, in part:
does Q have pos. density or at least positive
limsup in N?
"I think that one's easy. If q is missing only the primes
p_1, p_2, ..., p_n, then Q is missing only the numbers
not relatively prime to A = (p_1)(p_2)...(p_n), and you
can work out the density of those via the Euler phi-function."
Great, so from that is the density of Q wrt N =(a-1)*(b-1)*../(a*b*..) where
a, b,... are the finite
list of primes not in q? If so I think we can
calculate a sequence of densities of Q when
q is not cofinite--arrange the p_i in increasing
order and the kth density is the product
(p_1-1)*...(p_k-1)
divided by p1*..pk. Each density in this
sequence would be an upper bound for any
later density, and this sequence should
converge to the limit density. Is this
true?Thanks.
I may have mentioned earlier in this thread that the keyphrase here
is "sieve methods." There are books on the topic, and undoubtedly
web pages as well. Have a look.
--
Gerry Myerson (gerry@xxxxxxxxxxxxxxx) (i -> u for email)
.
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