"Brian M. Scott" <b.scott@[EMAIL PROTECTED]
> wrote in message
news:1pp3646txbjij.dnktxwl8ids9.dlg@[EMAIL PROTECTED]
> On Tue, 22 Jul 2008 20:32:18 +0100, Jack <jj@[EMAIL PROTECTED]
>
> wrote in <news:7Bqhk.147$WT.127@[EMAIL PROTECTED]
> in
> alt.algebra.help:
>
>>> Similarly, for large y the
>>> pro****tion of integers in [1, y] having only p as prime
>>> factor is about ln(y)/[y ln(p)]. The ratio of these
>>> pro****tions is ln(p)/ln(q), which is not 1.
>
>> Since 1/ln(p) converges asymptotically on 1/ln(q), ln(p)/ln(q) ---> 1.
>
> (1) In what you originally wrote, p and q were fixed,
> distinct primes, and y was increasing. The ratio
> ln(p)/ln(q) was therefore a fixed quantity different from 1.
>
> (2) What do you mean by '1/ln(p) converges asymptotically
> on 1/ln(q)'? In your original version you couldn't have
> meant a damned thing, since p and q were constants.
I never conceived them as such, and that was evident in my initial
reference
to p and q togther, "as y increases, each new prime, p, of value <sqrt(y)
is, on average, further apart in absolute terms than the previous (i.e.
the
prime that is adjacent but of lower value than p), but closer in
pro****tionate terms to the previous; and the value of 1/p, p in P gets
smaller and smaller; furthermore, for any p,q in P, where q < p and p and
q
are adjacent primes...."
I'm
> guessing that now you want to look at ln(p_n)/ln(p_{n+1}),
> where p_n is the n-th prime. That ratio may indeed approach
> 1 as n increases, but this has nothing to do with
> gobbledygook like '1/ln(p) converges asymptotically on
> 1/ln(q)'.
But what has it to do with my claim that ln(n) approaches
(o(1,a(P),P)*y)/a(P), where o(x,y,P) is the number of n in [x,y] for which
no member, p, of P divides n and a(P) is the product os all members of P?
Where you say
"I noted before that p(sqrt(y))/p(y) is about 2/sqrt(y) for
large y, so y * [1 - p(sqrt(y))/p(y)] is about y * [1 - 2/sqrt(y)] = y
*
[sqrt(y) - 2]/sqrt(y) = sqrt(y) * [sqrt(y) - 2], which is on the rough
order of y, not 1"
(the argument for which, as you know, I struggle to follow) I trust you
are
not suggesting that one of the two values ln(y) and (o(1,a(P),P)*y)/a(P)
is
greater than the other by a factor of approximately y.....?
Cheers.
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