On Mon, 21 Jul 2008 14:40:50 +0100, Jack <jj@[EMAIL PROTECTED]
>
wrote in <news:Ys0hk.20648$CE1.9987@[EMAIL PROTECTED]
> in
alt.algebra.help:
> "Brian M. Scott" <b.scott@[EMAIL PROTECTED]
> wrote in message
> news:ugkkzecxhz3c$.q5osvyzeigwb$.dlg@[EMAIL PROTECTED]
>> Apparently what you were really trying to say was something
>> like this, which is another matter altogether:
>> For any y in N let P(y) be the set of primes not
>> exceeding sqrt(y), and let n(y) be the number of
>> integers in [1, y] not divisible by any member of
>> P(y). Then for sufficiently large values of y,
>> n(y) is approximately p(y), the number of primes
>> in [1, y].
>> For any k in [1, y], either k is divisible by some prime in
>> P(y), or k is a prime greater than sqrt(y). Thus, n(y) is
>> simply the number of primes in [1, y] that are strictly
>> greater than sqrt(y), and hence n(y) = p(y) - p(sqrt(y)).
>> Now n(y)/p(y) = 1 - p(sqrt(y))/p(y). For large y we have
>> p(y) approximately equal to y/ln(y) and p(sqrt(y))
>> approximately equal to sqrt(y)/ln(sqrt(y)) =
>> 2 * sqrt(y)/ln(y), so p(sqrt(y))/p(y) is approximately
>> 2 * sqrt(y)/y = 2/sqrt(y), which does indeed tend to 0 as y
>> increases.
> But here there is no mention of the phi function.
Why should there be? There was no mention of it in your
original question, either. In fact, I was the one to
introduce it, in the context of providing a general
counterexample to William's interpretation of your original
claim. What happened to your interest in n(y)?
> I want to be certain that p(y)/y tends towards phi(m)
> \times y/(prod{p : p inP})).
Your terminology is incorrect: one does not normally speak
of a function f(y) tending towards another function g(y).
Do you mean that you want to be certain that the ratio of
the two function of y approaches 1 as y increases? That the
ratio lies between two fixed positive real numbers for all
sufficiently large y? Or some other notion of asymptotic
similarity?
Next, you haven't defined your notation. What is m? If
it's prod(P), why didn't you use the shorter notation in the
denominator as well as the numerator? If it's something
else, what is it? Is your P here the P(y) that I defined
above?
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